  We present the first fully general relativistic dynamical simulations of
Abelian Higgs cosmic strings using 3+1D numerical relativity. Focusing on
cosmic string loops, we show that they collapse due to their tension and can
either (i) unwind and disperse or (ii) form a black hole, depending on their
tension $G\mu$ and initial radius. We show that these results can be predicted
using an approximate formula derived using the hoop conjecture, and argue that
it is independent of field interactions. We extract the gravitational waveform
produced in the black hole formation case and show that it is dominated by the
$l=2$ and $m=0$ mode. We also compute the total gravitational wave energy
emitted during such a collapse, being $0.5\pm 0.2~ \%$ of the initial total
cosmic string loop mass, for a string tension of $G\mu=1.6\times 10^{-2}$ and
radius $R=100~M_{pl}^{-1}$. We use our results to put a bound on the production
rate of planar cosmic strings loops as $N \lesssim
10^{-2}~\mathrm{Gpc}^{-3}~\mathrm{yr}^{-1}$.
