Welch, 29, will arrive at Sandy Park from the Welsh region following previous playing stints with Cardiff Blues, Pontypridd and Cardiff RFC.
"We knew we'd have to sign a couple of second rows with the guys leaving - Peter Short, Bentos and Chad," coach Rob Baxter told the club website.
"We've been looking at Damian for some time, I think it's the third time I've spoken to him about potentially coming to Exeter.
"Initially I had a good look at him before he started playing professional rugby at Cardiff, but didn't quite think he was a fit for us at the time.
"He got a contract with the Cardiff Blues and pre him joining the Scarlets I had another chat with him but at the time he was thinking there were potentially some Wales caps there, he was playing very well.
"I've kept an eye on him all along and when I found out quite recently the Scarlets were prepared to release him from the second year of his contract we got onto things very quickly. He came down, did his testing again and he really is a great athlete."
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