We establish an approximate zero-one law for sentences of continuous logic over finite metric spaces of diameter at most $1$. More precisely, we axiomatize a complete metric theory $T_{\mathrm{as}}$ such that, given any sentence $σ$ in the language of pure metric spaces and any $ε>0$, the probability that the difference of the value of $σ$ in a random metric space of size $n$ and the value of $σ$ in any model of $T_{\mathrm{as}}$ is less than $ε$ approaches $1$ as $n$ approaches infinity. We also establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.