  We give a quantitative embedding of the Muckenhoupt class $A_1$ into
$A_\infty$. In particular, we show how $\epsilon$ depends on $[w]_{A_1}$ in the
inequality which characterizes $A_\infty$ weights: \[
  \frac{w(E)}{w(Q)} \leq \biggl( \frac{|E|}{|Q|} \biggr)^\epsilon, \] where $Q$
is any dyadic cube and $E$ is any subset of $Q$. This embedding yields a sharp
reverse-H\"older inequality as an easy corollary.
