  We calculate the leading-order matrix element for exclusive decays of $b \to
s\gamma$ in the quenched approximation of lattice QCD on a $24^3\times48$
lattice at $\beta=6.2$, using an O(a)-improved fermion action. The matrix
element is used to extract the on-shell form factor $T_1(q^2=0)$ for $B \to
K^*\gamma$ and $B_s \to \phi\gamma$, using two different assumptions about the
$q^2$ dependence of the form factors for these decays. For $B \to K^*\gamma$,
$T_1(q^2=0)$ is determined to be $0.159^{+34}_{-33}\pm 0.067$ or
$0.124^{+20}_{-18} \pm 0.022$ in the two cases. We find the results to be
consistent (in the Standard Model) with the CLEO experimental branching ratio
of $BR(B\to K^*\gamma) = (4.5 \pm 1.5 \pm 0.9) \times 10^{-5}$.
