Fix an integer $r \geq 3$. Given an integer $n$, we define $M_r(n)$ to be the set of metric spaces with underlying set ${1,\ldots,n}$ such that the distance between any two points lies in ${1,\ldots,r}$. We present results describing the approximate structure of these metric spaces when $n$ is large. As a consequence of these structural results in the case when $r$ is even, we obtain a first-order labeled $0$-$1$ law. This is joint work with Dhruv Mubayi.