  Photoproduction of $\rho^0$ mesons in ultra-peripheral Pb+Pb collisions has
been studied by the ALICE Collaboration at the CERN LHC. The strong photon flux
associated with relativistic charged nuclei leads to a very large cross section
for exclusive photoproduction of $\rho^0$ meson in interactions of the type $Pb
+ Pb \rightarrow Pb + Pb + \rho^0$. For a $\rho^0$ produced at mid-rapidity at
the LHC, the photon-nucleus center of mass energy is higher than in any
previous experiment.
  The ALICE detector is a general purpose detector dedicated to study
heavy--ion collisions. ALICE has excellent performance in the low $p_T$ region,
and can reconstruct charged particle tracks with 0.1 GeV/c $\leq p_T \leq 100$
GeV/c. In this analysis all tracks were required to be within ALICE's central
barrel. Analysis of data from the first heavy ion run at the LHC in 2010 will
be discussed in this paper.
