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.