Simulations performed with the particle-in-cell code Calder Circ show the feasibility of injection and acceleration of electrons in the laser wakefield created by few-femtosecond laser pulses with moderate energy at the few mJ level. A detailed study of the effect of the carrier-envelope phase of the pulse on the injection is presented. It is shown that using ionization injection with nitrogen as the target gas, the control of the optical phase allows production of high-quality and shot-to-shot stable electron beams. The electron bunches obtained have a relative energy spread of a few per cent, a bunch duration in the sub-fs domain, a divergence close to 10 mrad and a peak energy in the 10 MeV range, and could be produced in the near future at kHz repetition rates.