Betatron x-ray source from laser plasma interaction combines high brightness, few femtosecond duration and broad band energy spectrum. However, despite these unique features the Betatron source has a crippling drawback preventing its use for applications. Its properties significantly vary shot-to-shot and none of the developments performed so far resolved this problem. In this letter we present a simple method that allows to produce stable and bright Betatron x-ray beams. In addition, we demonstrate that this scheme provides polarized and easily tunable radiation. Experimental results show that the pointing stability is better than 10% of the beam divergence, with flux fluctuation of the order of 20% and a polarization degree reaching up to 80%