  We report the detection of high energy gamma-ray emission from the young and
energetic pulsar PSR B1509$-$58 and its pulsar wind nebula (PWN) in the
composite supernova remnant SNR G320.4-1.2 (aka MSH 15-52). Using 1 year of
survey data with the Fermi-Large Area Telescope (LAT), we detected pulsations
from PSR B1509-58 up to 1 GeV and extended gamma-ray emission above 1 GeV
spatially coincident with the PWN. The pulsar light curve presents two peaks
offset from the radio peak by phases 0.96 $\pm$ 0.01 and 0.33 $\pm$ 0.02. New
constraining upper limits on the pulsar emission are derived below 1 GeV and
confirm a severe spectral break at a few tens of MeV. The nebular spectrum in
the 1 - 100 GeV energy range is well described by a power-law with a spectral
index of (1.57 $\pm$ 0.17 $\pm$ 0.13) and a flux above 1 GeV of (2.91 $\pm$
0.79 $\pm$ 1.35) 10^{-9} cm^{-2} s^{-1}. The first errors represent the
statistical errors on the fit parameters, while the second ones are the
systematic uncertainties. The LAT spectrum of the nebula connects nicely with
Cherenkov observations, and indicates a spectral break between GeV and TeV
energies.
