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We show that noncommutative gauge theories with arbitrary compact gauge groupdefined by means of the Seiberg-Witten map have the same one-loop anomalies astheir commutative counterparts. This is done in two steps. By explicitlycalculating the $\epsilon^{\m_1\m_2\m_3\m_4}$ part of the renormalizedeffective action, we first find the would-be one-loop anomaly of the theory toall orders in the noncommutativity parameter $\theta^{\m\n}$. And secondly weisolate in the would-be anomaly radiative corrections which are not BRStrivial. This gives as the only true anomaly occurring in the theory thestandard Bardeen anomaly of commutative spacetime, which is set to zero by theusual anomaly cancellation condition.Comment: LaTeX 2e, no macros, no figures, 32 A4 page

Anomaly freedom in Seiberg-Witten noncommutative gauge theories