2002 Pontiac Grand Prix 3.1 liter. Acts like it has a dead battery.

At first I would try to start the car and it would do 1
rotation and then nothing not even a click. Now I go to
start it no rotation or click. I jump started the car and it
fired right up no issues. Let it run for a good hour Shut the
car off and try to start  immediately and  no click no
rotation just dead. I put a new battery in the car to see if
that was the problem hooked it up and tryed to start it and
the same out come as before. Not sure if it's the ignition
or starter.