  Superconducting micro-constrictions between Nb tips and high quality
MgB$_{2}$ pellets have been realized by means of a point-contact inset, driven
by a micrometric screw. Measurements of the current-voltage characteristics and
of the dynamical conductance versus bias have been performed in the temperature
range between 4.2 K and 500 K. Above the Nb critical temperature T$_{C}^{Nb}$,
the conductance of the MgB$_2$/normal-metal constrictions behaves as predicted
by the BTK model for low resistance contacts while high resistance junctions
show quasiparticle tunneling characteristics. Consistently, from the whole set
of data we infer the value $\Delta_{\pi} = 2.5 \pm 0.2$ meV for the
three-dimensional gap of MgB$_2$. Below T$_{C}^{Nb}$, low resistance contacts
show Josephson current and subharmonic gap structures (SGS), due to multiple
Andreev reflections. Simultaneous observations of both features, unambiguously
indicate coupling of the 3D band of MgB$_2$ with the Nb superconducting order
parameter. We found that the temperature dependence of the Josephson critical
current follows the classical Ambegaokar-Baratoff behavior with a value
$I_CR_N=(2.1 \pm 0.1)$ meV at low temperatures.
