1995 Oldsmobile Cutlass Car will not start

The battery in the car is good. I just replaced the starter today, but the problem won't go away. When I turn the key, nothing happens. When I let go of the key, it clicks twice underneath the hood somewhere. I've had this problem twice before, and in the past if I repeatedly turned the key, it would finally start right up. When it did make the connection, it started like usual, no problem. This time the car sits dead. I thought it was a starter problem, but that's not the case. It had been a very intermittent problem (only twice in the past 2 years). Any and all suggestions are more than welcome, thanks in advance for your help!