Go through the door at the top of the stairs and then through the second door. Wheatley will talk to you and tell you to find the Portal Gun on the podium in the middle of the room up ahead. Go over to it and the floor will collapse.
Up ahead you will see the actual Portal Gun on a podium that is surrounded by some murals. If you go close to the murals you can actually hear a faint voice. I don't know how that's even possible, but whatever.
There are some stairs that go up to the podium with the Portal Gun on it. Go up the stairs and walk onto the Portal Gun to get it. You can only make blue portals.
Make a blue portal on a wall that you can walk into. Walk through the blue portal. Then go up the metal ramp to the right.
Follow the path until the voice starts talking to you about lethal military androids. Make a blue portal on a wall and walk through it. Then turn to the left to see the exit door. Make a blue portal on the wall next to the exit and go back through the orange portal. Then go through the exit door, down the stairs and onto the elevator.