'98 Saturn, dashboard lights keep dimming in and out

I have a '98 Saturn SL2 with about 95,000 miles on it. I've recently started having some problems where the dashboard lights and electricity in general cuts in and out. The lights dim and then go back to normal. At one point, almost all the power cut out, the radio shut off, the gas meter went down to empty, then after 2 seconds it popped back up. I did some research online, and people with similar problems said a new alternator fixed the problem. Can anyone tell me what's wrong with my car? Thanks.