  Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation
field K. In a previous paper the authors togetehr with Q. Liu give various
characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve,
hyperelliptic genus 3 curve or bad) of the stable model of C: in terms of the
existence of a special plane quartic model and in terms of the valuations of
the Dixmier-Ohno invariants of C. The last one gives in particular an easy
computable criterion for the reduction type. However, it does not produce a
stable model, even in the case of good reduction. In this paper we give an
algorithm to obtain (an approximation of) the stable model when the reduction
of the latter is hyperelliptic and the characteristic of the residue field is
not 2. This is based on a new criterion giving the reduction type in terms of
the valuations of the theta constants of C. Some examples of the computation of
these models are given.
