1999 Pontiac Grand Prix won't start - unsure if it's battery

A couple of hours ago I started up the car as normal, then realized I'd left my phone in the restaurant. I turned off the car, saw the phone was in the backseat, then started the car right back up -- total elapsed time 3 seconds maybe. When starting up all I got was rapid clicking but the engine wouldn't turn over. We thought it was the battery (the lights were all dim, the radio went off, and the battery is due to be replaced in March 2017 anyway) so we got a jump from a tow truck, but it wouldn't turn over. As we attempted to start it during the jump I noticed the clicking stopped - there is now NO CLICKING at all when we try to start the car. The dash lights are basically normal brightness. What can I do to make absolutely sure it's not the battery or a battery cable? What could the problem be? Thanks.