  The process of exclusive elastic vector meson production in deep inelastic
scattering is investigated within the dipole model framework supplemented by
the small $x$ evolution. The dipole-proton amplitude is obtained from the
nonlinear Balitsky-Kovchegov evolution equation with impact parameter
dependence. This dipole amplitude is used to compute the differential cross
section for exclusive production of J/\Psi, \phi, and \rho, vector mesons.
These numerical calculations are compared with the wide range of experimental
data from HERA. Good agreement between the experimental data and the
calculations is found.
