Won't Start!!!

My car's been sitting for 15 months while I was in Iraq. I've replaced the battery, given her fresh gas, and checked everything I know how but she won't start. Won't even turn over. I turn the key, get a click and then nothing. It only clicks once and only on the 1st turn of the key. Have to wait about 15 minutes before it will even click again. I'm thinking bad starter, but that's not a lot to work with when it comes to my limited diagnostic abilities. You folks have any ideas? I'd love to hear them.
 
I know you said you replaced the battery, but it doesn't hurt to check the voltage anyway. It should be 12.8V fully charged. 12 - 12.5V is OK. Check it while turning the key to start. It shouldn't drop to less than 9V.

If tapping on the solenoid doesn't help, Bypass it with Jumper Cables as a test. Use just the red Jumper clamps. Go from the battery positive with one end and to the starter cable side of the solenoid with the other end. (Get ready for some sparks) If it cranks, it's your solenoid. If it doesn't, it's a bad starter.