  The magnetic penetration depth, $\lambda^{-2}(T)$, in the basal plane of a
magnesium diboride (MgB$_2$) film was measured using a two-coil mutual
inductance technique at 50 kHz. This film has $T_c\simeq 38$ K, $\Delta T_c
\leq 1$ K, and $\lambda(0)\sim 1500$ \AA. At low temperatures,
$\lambda^{-2}(T)$ shows a clear exponential temperature dependence, indicating
s-wave superconducting order parameter symmetry. However, the data are not
quantitatively well described by theory assuming a single gap. From the data
fit by the full BCS calculation assuming a double gap, the values of the two
distinct gaps were obtained: $\Delta_S(0)=2.61\pm0.41$ meV and
$\Delta_L(0)=6.50\pm0.33$ meV. The contributions of the small and the large
gaps to the total superfluid density at T=0 were estimated to be 21% and 79%,
respectively. Finally, we consider the effect of gap anisotropy on the
penetration depth measurements, and find that the anisotropy does not play a
significant role in determining the temperature dependence of the penetration
depth.
