  This paper continues a study initiated in [34], on the localization
transition of a lattice free field on $\mathbb Z^d$ interacting with a quenched
disordered substrate that acts on the interface when its height is close to
zero. The substrate has the tendency to localize or repel the interface at
different sites. A transition takes place when the average pinning potential
$h$ goes past a threshold $h_c$: from a delocalized phase $h<h_c$, where the
field is macroscopically repelled by the substrate to a localized one $h>h_c$
where the field sticks to the substrate. Our goal is to investigate the effect
of the presence of disorder on this phase transition. We focus on the two
dimensional case $(d=2)$ for which we had obtained so far only limited results.
We prove that the value of $h_c(\beta)$ is the same as for the annealed model,
for all values of $\beta$ and that in a neighborhood of $h_c$. Moreover we
prove that in contrast with the case $d\ge 3$ where the free energy has a
quadratic behavior near the critical point, the phase transition is of infinite
order $$\lim_{u\to 0+} \frac{ \log \mathrm{F}(\beta,h_c(\beta)+u)}{(\log u)}=
\infty.$$
