If in addition you assume that your domain is pseudoconvex then by a theorem of A.Aeppli what you want is true.The paper is titled :On the cohomology structure of Stein manifolds 1965 (Proc.Conf.Complex Analysis(Minneapolis Minn 1964) pages 58 to 70 Springer, Berlin. David's example indicates why one might need a hypothesis like pseudoconvexity.
Mohan Ramachandran
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