I disagree with Nos's assessment, with a caveat:
Assuming the rot hasn't progressed throughout the entire transom (which will mean digging out the affected wood until you reach clear wood), you can rebuild it with an epoxy resin (do a search for "transom repair" and you should find a good number of products suitable.) Addition of metal plates to help spread the load would probably be wise after this repair.
If your transom is painted wood, you'll probably have to come up with a mold or dam to hold the stuff in place until it cures. For a wood cored glass transom, simply pour the stuff in.
Then, redrill your engine mount bolt holes, and you're there.
If the entire transom is rotted, then I'll hop the fence to Nos's side.