To determine the superconducting gap function of YNi2B2C, the c-axis thermal conductivity kappa(zz) was measured in H rotated in various directions. The angular variation of kappa(zz) in H rotated within the ab plane shows a peculiar fourfold oscillation with narrow cusps. The amplitude of this fourfold oscillation becomes very small when H is rotated conically around the c axis with a tilt angle of 45 degrees. These results provide the first compelling evidence that the gap function has point nodes located along the a and b axes. This unprecedented gap structure challenges the current view on the pairing mechanism.