We have performed accurate measurements of the temperature dependence of the magnetic penetration depth of the new intermetallic superconductor MgB 2 using a high resolution single coil mutual inductance technique. Both sintered pellets and thin films exhibiting critical temperature values ranging between 37 and 38.4 K have been studied. In the case of the film, for T< T c/2 a clear exponential behavior is observed, which can be explained by a simple BCS s-wave model with a reduced value of the energy gap. On the contrary, pellets show no evidence of saturation, and the experimental results strictly follow a quadratic dependence down to the lowest temperatures. The experimental data on sintered samples can be explained in the framework of a conventional s-wave BCS theory using a two-band model.