  Elastic $\rho^0$ photoproduction has been measured using the ZEUS detector at
HERA. Untagged photoproduction events from $ep$ interactions were used to
measure the reaction $\gamma p \rightarrow \rho^0 p$ ($\rho^0 \rightarrow \pi^+
\pi^-$) at photon-proton centre-of-mass energies between 60 and 80~GeV and
$|t|<0.5$~GeV$^2$, where $t$ is the square of the four-momentum transferred at
the proton vertex. The differential cross section $d\sigma/dM_{\pi\pi}$, where
$M_{\pi\pi}$ is the invariant mass of the two pions, and the integrated cross
section, $\sigma_{\gamma p\rightarrow \rho^0 p}$, are presented; the latter was
measured to be $14.7\pm 0.4~(\mbox{stat.})~\pm2.4~(\mbox{syst.})~\mu\mbox{b}$.
The differential cross section $d\sigma/dt$ has an approximately exponential
shape; a fit of the type $A^{\prime}_{t} \exp{(-b^{\prime}_{t}|t| +
c^{\prime}_{t} t^2)}$ yields a $t$-slope $b^{\prime}_{t}=
9.9~\pm~1.2~(\mbox{stat.})~\pm 1.4~(\mbox{syst.})~\mu\mbox{b}$. The results,
when compared to low energy data, show a weak energy dependence of both
$\sigma_{\gamma p\rightarrow \rho^0 p}$ and of the $t$-slope. The $\rho^0$ is
produced predominantly with transverse polarisation, demonstrating that
$s$-channel helicity conservation holds at these energies.
