  We recently showed how it is possible to use a cubic Galileon action to
construct classical cosmological solutions that enter a contracting null energy
condition (NEC) violating phase, bounce at finite values of the scale factor
and exit into an expanding NEC-satisfying phase without encountering any
singularities or pathologies. A drawback of these examples is that singular
behavior is encountered at some time either just before or just after the
NEC-violating phase. In this Letter, we show that it is possible to circumvent
this problem by extending our method to actions that include the next order
${\cal L}_4$ Galileon interaction. Using this approach, we construct
non-singular classical bouncing cosmological solutions that are
non-pathological for all times.
