  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%
