"OK folks, now you can leave your bags here and go inside for your boarding passes," a cheery US army engineer tells the first arrivals for Amman. So we collect slips of paper that show no flight number, no seat number, no destination, not even a take-off time. There's a Burger King across the lot, but it's in a "high-security zone" which mere passengers cannot visit. There's no water for sale. There are so few seats that passengers stand in the heat outside what must be the biggest post office in the world, a vast US military sorting hanger with packets of mail for every one of the 146,000 troops in Iraq, standing 30ft high in racks.

But take a look at the passengers. There's a lady from the aid organisation Care heading off for a holiday in Thailand, and there's the Bishop of Basra in his black and red robes and dangling crucifix, and there's an outgoing television crew and the International Red Cross representative with a little Red Cross plane to catch to Kirkuk. There's also a British construction man up from Hilla who spent the previous night under fire with the local Polish battalion. "Rocket-propelled grenades and heavy rifle fire for two hours," he mutters. Of course, the occupation authorities never revealed that. Because things are getting better in Iraq.

Behind us, a series of giant four-engined jets are climbing in circles into the hot morning sky, big unmarked jobs that fly 180 degrees to the ground in tight circles to take off and land, so low you'd think they would trip the runway with their wing-tips - anything to avoid the ground-to-air missiles that America's enemies are now firing at aircraft in the "New Iraq". "It's routine," one of the American engineers confides to us. "We get shot at every night."

Among the other passengers, there's a humanitarian worker who's clearly had a nervous breakdown and some rather lordly Iraqi ladies escorted to check-in by an RAF officer with too much hair over his collar and, across the lot, a squad of American Special Forces soldiers enjoying the sun, heavy with black webbing, automatic rifles and pistols. Why do they all wear shades, I ask them? One of them takes off his sun-glasses. "What girl would look at us if they could see our real faces?" I agree. But they're an intelligent bunch of men, heavy with innuendo. Yes, they've got a safe house near Fallujah and combat casualties are sometimes "contained" within road accidents or drownings.

A guy called Chuck wants to confide in me. "You know the most precious resource about this country, Bob?" he asks. "It's the Iraqi people. There's a lot of protoplasm here." I was contemplating the definition of protoplasm when the first mortar came in, a thundering roar that had the passengers ducking like a theatrical chorus and a big white circle of smoke rising lazily from the other side of the runway. There's a whizzing noise and another clap of sound.

"They're getting better," Chuck tells me. "They must have put that one close to the runway." The other Special Forces lads nod approvingly. Another tremendous explosion, and they all nod together. Another big white ring rippling skywards, as if a giant cigar addict had sat down for a smoke by the runway. "Not bad at all," says Chuck's friend.

"We used to have a five-mile safety perimeter round the airport," Chuck says. "That's now down to two miles. The max anti-aircraft range is 8,000ft. So two miles is on the edge." Translation: US forces used to control five miles round the airport - too far to permit a man with a hand-held launcher to hit a plane. Ambushes and attacks on the Americans have reduced their control to a mere two miles. On the edge of that radius, a man might just hit a plane with a missile range of 8,000ft.

The Americans say there are two planes flying to Amman, at 10am and noon. Then another mortar round explodes in front of the hangars on the far side of the airport. And another.

"This," the Bishop of Basra sermonises to me, "is the continuation of our 22-year war." I call a colleague in Baghdad. Airport under mortar fire, I helpfully report. "Heard nothing about it, Bob," comes the reply. "How many mortars did you say?" But the Special Forces men are enjoying themselves. An Apache helicopter races over us to strafe the Iraqi guerrillas. "Some hope," says Chuck. "They've already pissed off." Technicians in guerrilla warfare, the Special Forces men are coolly appreciative of anyone's professionalism, including that of the enemy.

An American engineer pops up. If the TV crew will buy his guys Cokes, they can visit Burger King. A crackle of rifle fire from way beyond the airport perimeter. There must be a movie here, Walt Disney meets Vietnam.

The Airbus belongs, incredibly, to Royal Jordanian, the only international carrier to risk the run to Baghdad once a day. At the steps, there's a squad of Jordanian security men in white socks - Jordanian and Syrian plain-clothes cops always wear white socks - and they insist, right there on the runway, in checking over all our gear again. Computers turned on, computers turned off, cameras opened, closed, notebooks out, even a sheaf of readers' letters to be prowled over. The Apache flies back, rockets still in their pods.

Take-off is rather faster than usual. But there's no steady climb to cruising altitude. The Airbus turns sharply to port, G-forces pushing us into our seats, and there outside my window is the tented prison-camp city where the Americans keep more than 4,000 of their Iraqi prisoners without trial. The tents start to spin as the plane twists to starboard and then to port again, and there is the same prison camp outside my window, but this time upside down and turning anti-clockwise. I look around the cabin and notice fingers dug deep into arm-rests. The Airbus engines are howling, biting into the thinner air, and our eyes are searching for that thin trail of smoke that no one wants to see.

Then the pilot levels out. A Royal Jordanian stewardess in a bright white blouse arrives at our seats. Things are getting better in Iraq. "Juice or red wine, which would you like?" she asks me. Reader, which did I choose?