Update:
When you install the new battery, make sure that you don' try to solder directly to the battery. I was trying to play Silver recently, and the game save data was corrupted because the battery wasn't connected well enough. I tried a few different things to connect the battery better, but by the time I got the tab connected well to the battery, it had died from too much heat. The soldering iron can kill the battery it you heat it up too much.
I took the old battery out, and now I am trying a different approach to connect tabs to a new battery. I have made a kind of arc welder by taking apart a flash that you would put on top of a 35mm camera. I connected wires to the capaciter, so that when I touch both of them to the battery and tab, the electricity makes a flash, melting them together. Sometimes this works, sometimes this doesn't. I just need to find a place where I can get the tab to fuse firmly to the battery so I can put it in the game cartridge without losing a connection.