  Let $X$ be a proper algebraic variety over a non-archimedean, non-trivially
valued field. We show that the non-archimedean Monge-Amp\`ere measure of a
metric arising from a convex function on an open face of some skeleton of
$X^{\text{an}}$ is equal to the real Monge-Amp\`ere measure of that function up
to multiplication by a constant. As a consequence we obtain a regularity result
for solutions of the non-archimedean Monge-Amp\`ere problem on curves.
