[en] We show that noncommutative gauge theories with arbitrary compact gauge group defined by means of the Seiberg-Witten map have the same one-loop anomalies as their commutative counterparts. This is done in two steps. By explicitly calculating the ε^μ₁^μ₂^μ₃^μ₄ part of the renormalized effective action, we first find the would-be one-loop anomaly of the theory to all orders in the noncommutativity parameter θ^μν. And secondly we isolate in the would-be anomaly radiative corrections which are not BRS trivial. This gives as the only true anomaly occurring in the theory the standard Bardeen anomaly of commutative spacetime, which is set to zero by the usual anomaly cancellation condition. (author)