The 19-year-old, who had been linked with a New Year move to Morecambe, has impressed in 16 Welsh Premier appearances since arriving at Maesydre.
Cudworth played in the Preston North End youth set-up and reserves before being released last summer.
"He's a great prospect and a very good signing for us," Rhyl player-manager Greg Strong told Lilywhites Online.
Former Airbus UK keeper Phil Palethorpe is reported to be the replacement for Cudworth at Welshpool.
Palethorpe, who joined Chester City from Tranmere Rovers in season 2006/7, made 27 Welsh Premier appearances for the Wingmakers last season and has also spent time on-loan at Tamworth.