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."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
$29.99
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.
