Northern Ireland goalkeeper Carson needs surgery after suffering a serious knee injury and will be out for "a minimum of three months".
Archer, 27, has signed a short-term deal until January, having been without a club since leaving Fulham at the end of last season.
"Jordan comes with a strong pedigree," said manager Stephen Robinson.
"To be able to bring in a 'keeper of Jordan's quality is fortunate and he will challenge Aaron Chapman for the jersey."
Former Tottenham and Millwall goalkeeper Archer, whose grandparents are from Clydebank, earned his sole Scotland cap in the friendly with Peru in May 2018.
Prior to that, he made 23 appearances for Scotland at various age groups.
"I'm glad to be here, and I'm ready if called upon," Archer said.
"It's my first time playing club football in Scotland, and I am looking forward to the new challenge."
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