  $^{11}B$ NMR and nuclear spin-lattice relaxation rate (NSLR) are reported at
7.2 Tesla and 1.4 Tesla in powder samples of the intermetallic compound $MgB_2$
with superconducting transition temperature in zero field $T_c$ = 39.2 K. From
the first order quadrupole perturbed NMR specrum a quadrupole coupling
frequency of 835 $\pm$ 5 kHz is obtained. The Knight shift is very small and it
decreases to zero in the superconducting phase. The NSLR follows a linear law
with $T_1T$ = 165 $\pm$ 10 (sec K) . The results in the normal phase indicate a
negligible $s$-character of the wave function of the conduction electrons at
the Fermi level. Below $T_c$ the NSLR is strongly field dependent indicating
the presence of an important contribution related to the density and the
thermal motion of flux lines. No coherence peak is observed at the lower field
investigated (1.4 T).
