  The zeros of the size-$n$ partition functions for a statistical mechanical
model can be used to help understand the critical behaviour of the model as
$n\to\infty$. Here we use weighted Dyck paths as a simple model of
two-dimensional polymer adsorption, and study the behaviour of the partition
function zeros, particularly in the thermodynamic limit. The exact solvability
of the model allows for a precise calculation of the locus of the zeros and the
way in which an edge-singularity on the positive real axis is formed. We also
show that in the limit $n\to\infty$ the zeros converge on a lima\c{c}on in the
complex plane.
