  Using Scanning Tunneling Microscope at low temperature we explore the
superconducting phase diagram in the $\pi$-band of the two-band superconductor
MgB$_2$. In this band the peculiar shape of the local tunneling spectra and
their dynamics in the magnetic field reveal the complex character of the
quasiparticle density of states (DOS). The gap in the DOS is first rapidly
filled with states in raising the magnetic field up to 0.5 T and then slowly
approaches the normal state value : The gap is observed up to 2 T. Such a
change in the DOS dynamics suggests the existence of two terms in the DOS of
the $\pi$-band: a first one, reflecting an intrinsic superconductivity in the
band and a second one, originating from an inter-band coupling to the
$\sigma$-band. Our findings allow a deeper understanding of the unique phase
diagram of MgB$_2$.
