The Coatbridge-born shot-stopper, a representative of Scotland at under-21 level, joined the Magpies from Scottish League Two outfit East Fife in August 2022 following an impressive trial.
The 21-year-old, who began his career in Celtic's youth ranks, moves to Brunton Park on a two-year deal and will link up with the Cumbrians on Monday, 1st July following his expiration of his contract at St. James' Park.
Speaking on Smith's arrival, Carlisle manager Paul Simpson told carlisleunited.co.uk: "Jude is a goalkeeper we've had our eye on for a while. He was sought after when he left Scotland as a young lad but Newcastle won the race for him.
"We've followed his progress since then so we're really pleased to get him in the building now."
Everybody at the club would like to wish Jude the best of luck for the future.