Let Ω = Πi ∈ IUi be an arbitrary product of planar domains {Ui}i ∈ I of some type, Ω endowed with the product topology. We consider A(Ω) the Banach algebra of all functions f from the closure of Ω to C that are continuous on the closure of Ω and separately holomorphic in Ω. We show that finite sums of finite products of rational functions of one variable with prescribed poles off the closure of Ui are uniformly dense in A(Ω).