  A study of the pressure effect on the magnetic penetration depth $\lambda$ in
polycrystalline MgB$_{2}$ was performed by measuring the temperature dependence
of the magnetization under an applied pressure of 0.15 and 1.13 GPa. We found
that $\lambda^{-2}$ at low temperature is only slightly affected by pressure
[$\frac{\Delta \lambda^{-2}}{\lambda^{-2}} = 1.5(9)%$], in contrast to cuprate
superconductors, where, in the same range of pressure, a very large effect on
$\lambda^{-2}$ was found. Theoretical estimates indicate that most of the
pressure effect on $\lambda^{-2}$ in MgB$_2$ arises from the electron-phonon
interaction.
