Battery woes - help please?

I got in the car after work, and the car wouldn't crank at all. The first thing I checked was to see if I left the lights on, but I hadn't. The battery is about 3 years old. A coworker gave me a jump. When I got home I turned it off, and tried to turn it back on, but it was immediately dead again. The battery is reading 13 volts by my meter. There was some corrosion at the terminals so I cleaned them up real good, but still no luck. Can this still be a dead battery showing 13 volts? I still have interior lights and stuff, just not enough to crank the engine.