I don't know that Sony any longer has any facilities for repairing a T1. Batteries can be purchased fairly cheaply, but replacing them requires de-soldering and re-soldering three wires.
https://www.youtube.com/watch?v=mQovQSrbpPo
Battery available on eBay (if in the U.S.) ...
https://www.ebay.com/itm/Battery-for...0AAOSw4GFbLe7H
You can also usually find cheap T1s on Shop Goodwill or eBay. (Not much there right now.) It might not be a bad time to upgrade to a T2.