The 41-year-old moves to the County ground after serving as bowling coach and assistant coach at Warwickshire, having begun his coaching career overseeing the bowling at Essex in 2008.
Welch also spent six years as a player at Derbyshire, taking over as captain when Luke Sutton left for Lancashire in 2006.
"My experiences with Essex and Warwickshire have been a fantastic grounding, allowing me to learn my trade as a coach," he said.
"I'm thankful for those opportunities which have fuelled my ambition to work at the top of the coaching tree.
"The opportunity to do that back at Derbyshire, where I had my best years as a player, is perfect for me.
"The new Elite Cricket Performance structure shows the Club's ambition and I completely buy into their vision and share the desire for success."
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