The goalkeeper will make his Quakers debut in Saturday's game at Durham City and replaces Jack Norton, who has been released.
Bell was West's first-choice keeper and played against Quakers when Martin Gray's side lost 2-1 at Darlington Road last month.
The keeper also played in the FA Vase Final at Wembley last season, as did Terry Galbraith, for Dunston UTS, who Quakers signed a fortnight ago.
"He's very well thought of in the Northern League and I'm absolutely delighted to sign him," said Gray. "With Craig Turns in the squad as well, we've got two very good keepers."
Norton began the season as Darlington's first-choice keeper before briefly losing his place to Turns.
However, Turns suffering a shoulder injury two weeks ago meant Norton was handed a reprieve, playing in last Saturday's match at Newton Aycliffe.
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