  We study the horocycle flow on the stratum of translation surfaces
$\mathcal{H}(2)$. We show that there is a sequence of horocycle ergodic
measures, each supported on a periodic horocycle orbit, which weakly converges
to an invariant, but non-ergodic, measure by $\mathrm{SL}_2(\mathbb{R})$. As a
consequence, we show that there are points in $\mathcal{H}(2)$ whose horocycle
flow orbits do not equidistribute towards any invariant measure.
