Quoted: I would bet my next paycheck on this:
Lower heating element. Easy to replace - about $20-$30
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Yup. I had the exact same symptoms, and my lower heating element had gone out. You didn't mention whether you have a gas or electric water heater, but I'll bet it's electric, and the lower element is dead.
The upper element is keeping the water in the top of the tank hot, which is what is drawn off first, but the lower element isn't working, so the water in the bottom of the tank is cold.
If you have a continuity test meter or a multimeter, you can easily check the element. Turn off the power to the heater and remove the cover plate over the lower element. if there's any insulation or a plastic cover plate, remove them, and then touch one probe of the tester to each screw/bolt on the surface of the heating element (it'll be the 2-inch wide bolt with two screws on the surface of it). If you have no continuity, the element is bad.
Drain the tank and unscrew the element with a friggin huge socket if you happen to have one, or buy a cheapo element removal tool from the hardware store (5 bucks max). Unhook the wires on the element and unscrew it to remove it. Replace it with a new element and hook the wires back up. Refill the tank and check for leaks. If it's good, replace the plastic guard and insulation, put the cover back on, and power on the heater. Enjoy longer showers, and feel good that you didn't have to buy a new heater.
ETA: Well, crap. I see that your water heater is gas. Nevermind.