If you took the time to register Lightroom with Adobe when you first installed it then you can log into your account and find the serial number there. If you are like most people, and didn't bother to register, then you will have to contact Adobe to see what can be done.
Lightroom 1.0 is terribly obsolete. If you are using a camera that shoots raw images it won't be supported in that version of Lightroom unless the camera itself is really old. Lightroom has changed a lot since it was first introduced.
Thanks for replying, Jim. I wasn't planning to use it for raw images. I shoot film, black and white, and have many negatives I want to print. I am unfamiliar with the upgrades in Lightroom. I wasn't unhappy with the version I have. It was useful for my purposes.
Because it was a few years ago, I don't remember if I registered the product. I was able to login to my account. But could not find a path to communicate with Adobe. I will keep trying.