The first person to notice something wrong is the Iraqi translator. Morning prayers should begin at 5:35 am in Baghdad on this particular date in May. Why, he wonders, is a muezzin in the Ghazaliya district calling the faithful at 5:20, fifteen minutes too soon? He tells his platoon leader, Lieutenant Matthew Holtzendorff, about his suspicion. The early call to prayer might be a previously-arranged signal to warn that the Americans are on the move.

Holtzendorff, 36, short and blond, has been stationed in Ghazaliya for more than half a year. He's led dozens of patrols through the neighborhoods. He knows every street, and on some streets he knows every house. He's especially familiar with Sector 651, which the Iraqis call Kafaat. The second house, which he orders his men to search, is a hit. Half-full glasses are still standing on the living room table, and two mopeds are parked in the front yard. The insurgents must have just fled on foot.

They've left a video camera on one of the mopeds. When Holtzendorff switches it on, the camera shows images of a gun battle, an attempted murder, perhaps taken from the roof of a building. But which roof?

Ghazaliya was once a well-kept neighborhood for Saddam's officers, professors and diplomats in Baghdad's extreme west. It was the first neighborhood to be confessionally "cleansed" by insurgents who had seeped into the neighborhood from the Sunni triangle. A black X has been scrawled on the garden wall of every other house -- a sign that a Shiite or otherwise unwanted family had been driven out.

The extremists' brutality in Ghazaliya was unprecedented. Men who had been beheaded or shot through the eye or mouth lay in the gutter. Neither the police nor the Iraqi army dared come to the neighborhood to collect the dead. Residents, and not just Shiites, left the neighborhood by the thousands.

The US military had largely withdrawn from Baghdad's residential areas, but when the American-Iraqi security plan took effect last February -- mapped out by the new US commander in Iraq, General David Petraeus -- it meant a return of troops to the city. More troops, and joint patrols with Iraqi army units. It also meant daytime and nighttime patrols to pressure the insurgents.

Holtzendorff and his company leader, Captain Jon Brooks, 28, will spend the morning with the Iraqi translator in front of a computer, analyzing voices on the video. The house they're looking for must be in southeast Ghazaliya, with an unobstructed view of the Abu Ghraib highway. The evening patrol will take a closer look at the house, and Brooks will lead the patrol.

From Cesspool to Walled Community

For about a year, the violence in Ghazaliya was so bad that no merchants dared to show themselves on the street. But in mid-March the Nafla market in the center of the neighborhood reopened. A heavy stench still lingers over the kebab and vegetable stands, but women, men and children are shopping here again.

Ghazaliya is a cesspool, a garbage dump covering two square kilometers. And yet there are streets where boys play soccer again and mothers push baby carriages. Ruins abut villas where the hedges are neatly trimmed and the lawns mowed. Ghazaliya today looks like a blend of Grozny and Beverly Hills.

As part of the crackdown in mid-February, the Second Battalion of the 12th Cavalry Regiment of the First Infantry Division requisitioned two villas on the western edge of Ghazaliya. Soon stacks of bodies were being dumped on the empty lots in front of the houses. The new outpost, which now amount to a small fortress, was named "Thrasher" in honor of Sergeant Robert Thrasher, who was killed a block away on Feb. 11.

"One day," says Lieutenant Mike Obal, 23, talking about the barbaric conditions before Thrasher was established, "we brought up five bodies from the Alpha road to a new Iraqi police station, when someone sent us a radio message to tell us to pick up yet another body."

Obal's men found an old woman at the same spot where they had just picked up the other bodies. Her body was still warm and her shopping bag lay on the ground next to her. The terrorists had shot a passerby at random, "just like that, to lure us back," says Obal. Under the street, the insurgents had also hidden a 200-liter oxygen bottle filled with explosives, which detonated when a heavy tank later drove over the site, tearing a plate-sized hole in the tank's armor and fracturing the driver's hand, which was on the control stick.

Today, a wall separates the Sunni Ghazaliya neighborhood from the mixed northern section and the villages to the west. "Thrasher" and two other outposts are manned by about 100 soldiers each, and an entire battalion of the Iraqi army has moved into Ghazaliya. The district, still home to about 15,000 people, is now being guarded by a total of 800 soldiers -- one for every 20 residents. This is almost exactly the ratio that cautious US military officials believed, just before the invasion in 2003, would be necessary to stabilize Iraq.

Maintaining the Status Quo

Dr. Kamal Z., 64, who studied in Leipzig in the 1960s, says he's replaced the glass in his living room window four times in recent months. "That damned al-Qaida!" the white-haired man exclaims. "And those damned explosions!" A small new generator hums in his front yard, and the TV set in his bedroom is tuned to Deutsche Welle.

"Believe me, Lieutenant," he says to Holtzendorff, who knocks on the Iraqi's door with his patrol, "at the moment none of us wants to know exactly who our neighbors are, or what they are doing. Wouldn't you like a little glass of cognac?" he asks, waving a small bottle of Rémy Martin.

Dr. Z. knows Hannelore K. well -- the German woman who was kidnapped with her son in Ghazaliya in February and seems to have vanished without a trace. Lieutenant Holtzendorff, sitting on the sofa in full military gear, also knows who Ms. K. is. His unit received a tip about her in mid-March. Several masked men had pulled two blindfolded people into a house north of Sadik Street -- presumably Hannelore and her son. Using night vision equipment Holtzendorff's team had searched several houses in the street. "It looks like we got the color of the garage door wrong at the first house we entered," he says. When they entered the next house, the kidnappers had apparently escaped through a back door.

But where exactly are the insurgents who were forced out of Ghazaliya? Many are still there, as evidenced by the roadside bombs that detonate every day. But where are the others?

At five in the afternoon, Holtzendorff and Captain Brooks, with some 20 men, discover the house where the video found that morning was recorded. The villa has been abandoned for months. The soldiers shoot open two large padlocks and then search for hidden weapons and ammunition between old pieces of furniture, in toilet tanks, and underneath loose bathroom tiles for hidden weapons and ammunition. They peer into the water tank on the roof and calculate -- based on fresh bullet holes on the balcony -- the direction from which shots were last fired at the house. A neighbor tells the soldiers that he noticed a black Kia with a scratched license plate. The vehicle, he says, is not from the neighborhood.

A few shots are fired, possibly from the other side of the highway. Brooks and Holtzendorff lean against a railing on the roof, gazing across at Baghdad's former horse track. The track is in the Khadraa neighborhood, where conditions today are similar to conditions in Ghazaliya three months ago. It is quite possible that the black Kia is already driving around in Khadraa now, and it is also quite possible that the German hostages are being held in a basement there.

In the slow, confusing street campaign to secure Baghdad, what constitutes a military success? How do you know when you've won?

"We certainly don't measure it based on the number of insurgents we arrest or kill," says Major James Orr, 45, of the Military Transition Team, which coordinates the efforts of American and Iraqi soldiers in Ghazaliya. "What we care about is that the Iraqi army can continue to maintain, on its own, the status quo we have reached by today."