  We show that, given an absolutely continuous horizontal curve $\gamma$ in the
Heisenberg group, there is a $C^1$ horizontal curve $\Gamma$ such that
$\Gamma=\gamma$ and $\Gamma'=\gamma'$ outside a set of small measure.
Conversely, we construct an absolutely continuous horizontal curve in the Engel
group with no $C^1$ horizontal approximation.
