  Magnetic relaxation in a MgB$_2$ superconductor was measured. The temperature
dependence of the normalized relaxation rate was determined for three different
magnetic fields. By extrapolating these rates to T=0 K, we find that these
extrapolations do not approach zero, indicating quantum tunneling of vortices
in the superconductor. A quantum correction of the relaxation rate, followed by
the correction of the magnetic moment, is proposed. Using the quantum
correction, we find that $U_0$ increases with decreasing temperature and
approaches a maximum at T=0.
