  Given a connected graph $G=(V,E)$ and a length function $\ell:E\to {\mathbb
R}$ we let $d_{v,w}$ denote the shortest distance between vertex $v$ and vertex
$w$. A $t$-spanner is a subset $E'\subseteq E$ such that if $d'_{v,w}$ denotes
shortest distances in the subgraph $G'=(V,E')$ then $d'_{v,w}\leq t d_{v,w}$
for all $v,w\in V$. We show that for a large class of graphs with suitable
degree and expansion properties with independent exponential mean one edge
lengths, there is w.h.p.~a 1-spanner that uses $\approx \frac12n\log n$ edges
and that this is best possible. In particular, our result applies to the random
graphs $G_{n,p}$ for $np\gg \log n$.
