Smith, who has been on trial at the Greenhous Meadow for the past couple of weeks, impressed enough to persuade manager Graham Turner to tie him down permanently.
The Shrews boss told Town's official website: "He is a good character and wants to play for the club and help to push us on to success. It's good news that he has chosen to join the club and I look forward to continuing the hard work with him on the training field.
"With Ben joining that gives us good competition in the goalkeeping position, with him pushing Chris Neal and Hakan Duyan all the way."