  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 $\sigma$ in the language of pure metric spaces and any $\epsilon>0$,
the probability that the difference of the value of $\sigma$ in a random metric
space of size $n$ and the value of $\sigma$ in any model of $T_{\mathrm{as}}$
is less than $\epsilon$ approaches $1$ as $n$ approaches infinity. We also
establish some model-theoretic properties of the theory $T_{\mathrm{as}}$.
