1990 Buick Lesabre won't start

I have a friend who has a 1990 Lesabre. 3.8L motor. He went to crank it and it would only click. Thinking it was a bad battery he took another battery (from another car) and put in it. It fired right up. He went and bought a new battery and put it in and it clicks again and won't start at all. He says all the lights do work. Any suggestions on what it may be and ways to test to see if a certain part is the problem without buying the part. Thanks for your help.
Bethany

Bad battery cable connections, bad ground, bad starter solenoid, all possibilities that need to be checked out.