We investigate an ionization injection scheme in a “weakly” non-linear regime of a wakefield, driven by sub-TW, few-cycle laser pulses in a single-stage, high-Z gas. This medium simultaneously provides the background wake fluid electrons from its lower ionization states and the necessary dephased electrons from its higher ionization states. Two dimensional-particle-in-cell simulations show the generation of relativistic electron beamlets having up to 15 MeV peak energy, with a narrow energy-spread and sub-fs duration. Since the currently-available sub-TW, few-cycle laser systems operate at kHz repetition rates, the presented scheme is capable of producing kHz attosecond electron bunches and their associated radiations which can find unique applications, for instance, in attosecond diffraction and microscopy.