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This paper presents a noise cancellation method based on the ability to efficiently cancel a close target speaker contribution from the signals observed at a microphone array. The proposed method exploits this specificity in the case of the hands-free speech interface. This method is in particular able to deal with non-stationary noise. The method can be divided in three steps. First, the steering vector pointing at the target user is estimated from the covariance of the observed signals. Then the noise estimate is obtained by cancelling the user's contribution. During this step the speech pauses are also estimated. Finally a post-filter is used to suppress this estimated noise from the observed signals. The post-filter strength is controlled by using the estimated noise during the speech pauses as reference. A 20k-words dictation task in presence of non-stationary diffuse background noise at different SNR levels illustrates the effectiveness of the proposed method.