Cork, son of former England international Dominic, has made his breakthrough to the first team this year after impressing for the second XI.
The 19-year-old all-rounder, a product of the Derbyshire Academy, took six wickets in as many NatWest T20 Blast matches for the Falcons this season.
Elite performance director Graeme Welch said: “We’re delighted the Academy is bringing through talented players who are worthy of professional deals, of which Greg is the latest and more will certainly follow.
“We’ve thrown him in at the deep end in Twenty20 cricket this summer and his performances have been improving with each game.
“He’s left-arm over which is a commodity in itself, we like his variation and the way he can mix it up and there’s also plenty of potential there with the bat.
“Having graduated to the professional ranks on a summer contract this season, Greg has never looked out of place.
“If he continues to work hard at his game, listens and continues learning his trade, there’s potential there for a really bright future.
“Our coaching structure here – and no doubt some words of wisdom from his dad Dominic – will ensure that all the advice and support is available for Greg to succeed as a professional player.”
Meanwhile, the club will not be handing Borrington a new contract and the opening batsman has left the 3aaa County Ground with immediate effect.
The 26-year-old, who played 49 first-class matches for the county, is now looking for new opportunities elsewhere.
Welch said: “Paul has been with the club for a long time and is a popular player among the lads, so it’s sad to see him go.
“However, we have to move on and I speak for everyone at the club when I wish the lad all the very best for the future.”