  We demonstrate that a long-propagating plasma bubble executing undulatory
motion can be produced in the wake of two co-propagating laser pulses: a
near-single-cycle injector and a multi-cycle driver. When the undulation
amplitude exceeds the analytically derived threshold, highly-localized
injections of plasma electrons into the bubble are followed by their
long-distance acceleration. While the locations of the injection regions are
controlled by the carrier-envelope phase (CEP) of the injector pulse, the
mono-energetic spectrum of the accelerated sub-femtosecond high-charge electron
bunches is shown to be nearly CEP-independent
