  We analyze the collective surface diffusion coefficient, $D_c$, near a
first-order phase transition at which two phases coexist and the surface
coverage, $\te$, drops from one single-phase value, $\te_+$, to the other one,
$\te_-$. Contrary to other studies, we consider the temperatures that are
sufficiently sub-critical. Using the local equilibrium approximation, we
obtain, both numerically and analytically, the dependence of $D_c$ on the
coverage and system size, $N$, near such a transition. In the two-phase regime,
when $\te$ ranges between $\te_-$ and $\te_+$, the diffusion coefficient
behaves as a sum of two hyperbolas, $D_c \approx A/N|\te - \te_-| + B/N|\te -
\te_+|$. The steep hyperbolic increase in $D_c$ near $\te_\pm$ rapidly slows
down when the system gets from the two-phase regime to either of the
single-phase regimes (when $\te$ gets below $\te_-$ or above $\te_+$), where it
approaches a finite value. The crossover behavior of $D_c$ between the
two-phase and single-phase regimes is described by a rather complex formula
involving the Lambert function. We consider a lattice-gas model on a triangular
lattice to illustrate these general results, applying them to four specific
examples of transitions exhibited by the model.
