  The construction introduced by Gross, Hacking and Keel allows one to
construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth
rational projective surface and $D$ a certain type of Weil divisor supporting
an ample or anti-ample class. In that paper they proved two convergence
results. Firstly that if the intersection matrix of $D$ is not negative
semi-definite then the family they construct lifts to an algebraic family.
Secondly they prove that if the intersection matrix is negative definite then
their construction lifts along certain analytic strata on the base, and then
over a formal neighbourhood of this. In the original version of that paper they
claimed that if the intersection matrix were negative semi-definite then family
in fact extends over an analytic neighbourhood of the origin but gave an
incorrect proof.
  In this paper we correct this error. We explain how the general Gross-Siebert
program can be used to reduce construction of the mirror to such a surface to
calculating certain relative Gromov-Witten invariants. We then relate these
invariants to the invariants of a new space where we can find explicit formulae
for the invariants. From this we deduce analytic convergence of the mirror
family, at least when the original surface has an $I_4$ fibre.
