If in addition if 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.