Dead key fob! Kind of stinks -- especially since my 2002 Ford Escape only has one door with an external lock. I did some looking online to find out how to replace the battery and reprogram it for my car. Here are the steps I followed:

The battery
Remove the keychain loop from the fob. Pop open the fob. I used a flat head screwdriver, but I have read that a dime works also. Just insert the screwdriver into the little slot inside the keychain mount and twist it (gently!). Make sure it goes into the main body of the key fob -- there should be no pressure applied to the keychain mount itself since that plastic is very weak. Here is the opened fob. It has a nice rubber seal around the inside. Instructions are printed inside. Interesting, the battery is a CR2032, also known as a "CMOS battery" for PC computer motherboards.
I've been starting to see these all over in other applications -- even those music-playing greeting cards use them.
If you have one of those cards laying around, this might be a good way to recycle that battery. My digital multitester shows it is only putting out 2.93 volts. I guess that must not be enough(!). My replacement battery is 3.3 volts. So I put in the replacement and snap the fob back together with my fingers.

Reprogramming the fob
I found this procedure rather non-intuitive. I'd like to thank these guys for their help. I have read that this procedure works on a lot of Ford trucks, your mileage may vary. Note that I did this with the car doors shut. I'm not sure if that matters. I also only had one fob.