Sounds to me like you have a short between the ignition switch and the starter solenoid. So start by thr process of elimination. Most alarm systems have a starter disable feature. Since the alarm is on the fritz you may first want to chase down the ignition kill wire from the alarm box to the wire it's tapped into and remove it.

If that doesn't fix it then check the cable between the battery and the starter solenoid for and corrosion. Next check the battery by doing a gravity test on the eletrolyte in each cell. If all is OK there then consider drinking a beer. You are most likely going to be unbolting that starter to take to the autoparts store. Good luck!
<img src="/forums/images/graemlins/patriot.gif" alt="" />