2001 Honda Civic Air conditioner stops working after ten minutes

2001 Honda Civic mileage: 75,000. My air conditioning works fine first time you turn it on. Runs for ten minutes and then stops cooling. I had it checked out and there were no leaks in the system. The mechanic's guess was that the compressor doesn't turn on when it is hot. He gave me an estimate of $1100.00 to replace the compressor, receiver dryer, evacuate, recharge etc. Is this the only way to fix the problem? Could it be something else that causes compressor to stop when it is hot? I would appreciate it a lot if someone could help me out.