1999 Buick Century fuel pump problems?

Hello. A few months ago I, for the first time ever, ran out of gas. I went to get gas to put in but the car would try to start but would sputter and eventually shut off. After several hours it would start back up.
It seemed when the fuel got low it would happen and the same thing, a few hours later it would start.
I was told it was my fuel pump. I did notice a big difference in the noise of the pump when it started fine and when it didn't I couoldn't hear the pump working. My question is, before I spend alot of money on a fuel pump, Do you believe that this is the actual problem?
I appreciate any help you may give me. Have a good day,
tom