On projective modules over polynomial rings

We show that if A is a domain obtained by a certain rather general type of pullback from Prufer and Bezout domains, then it satisfies the following property: for every n greater than or equal to 0, all finitely generated projective A[X-1,...,X-n]-modules are extended from A. We construct a wide class of domains of Krull dimension one and arbitrary valuative dimension that satisfy the above property.