  The exclusive production of $\rho^0$ mesons in deep inelastic electron-proton
scattering has been studied using the ZEUS detector. Cross sections have been
measured in the range $7 < Q^2 < 25$ GeV$^2$ for $\gamma^*p$ centre of mass
(c.m.) energies from 40 to 130 GeV. The $\gamma^*p \to \rho^0 p$ cross section
exhibits a $Q^{-(4.2 \pm 0.8 ^{+1.4}_{-0.5})}$ dependence and both
longitudinally and transversely polarised $\rho^0$'s are observed. The
$\gamma^*p \to \rho^0 p$ cross section rises strongly with increasing c.m.
energy, when compared with NMC data at lower energy, which cannot be explained
by production through soft pomeron exchange. The data are compared with
perturbative QCD calculations where the rise in the cross section reflects the
increase in the gluon density at low $x$. the gluon density at low $x$.
