With high resolution (0.25 x 0.18) ALMA CO 3-2 observations of the nearby (D=21 Mpc, 1=102 pc), extremely radio-quiet galaxy NGC1377, we have discovered a high-velocity, very collimated nuclear outflow which we interpret as a molecular jet with a projected length of +-150 pc. Along the jet axis we find strong velocity reversals where the projected velocity swings from -150 km/s to +150 km/s. A simple model of a molecular jet precessing around an axis close to the plane of the sky can reproduce the observations. The velocity of the outflowing gas is difficult to constrain due to the velocity reversals but we estimate it to be between 240 and 850 km/s and the jet to precess with a period P=0.3-1.1 Myr. The CO emission is clumpy along the jet and the total molecular mass in the high-velocity (+-(60 to 150 km/s)) gas lies between 2e6 Msun (light jet) and 2e7 Msun (massive jet). There is also CO emission extending along the minor axis of NGC1377. It holds >40% of the flux in NGC1377 and may be a slower, wide-angle molecular outflow which is partially entrained by the molecular jet. We discuss the driving mechanism of the molecular jet and suggest that it is either powered by a very faint radio jet or by an accretion disk-wind similar to those found towards protostars. The nucleus of NGC1377 harbours intense embedded activity and we detect emission from vibrationally excited HCN J=4-3 v_2=1f which is consistent with hot gas and dust. We find large columns of H2 in the centre of NGC1377 which may be a sign of a high rate of recent gas infall. The dynamical age of the molecular jet is short (<1 Myr), which could imply that it is young and consistent with the notion that NGC1377 is caught in a transient phase of its evolution. However, further studies are required to determine the age of the molecular jet, its mass and the role it is playing in the growth of the nucleus of NGC1377.