  Young supernova remnants are thought to be the sites where cosmic ray
acceleration occurs by the mechanism of diffusive shock acceleration. The
maximum energy gained in this process is conventionally extimated to have a
value close to, but distinctly below, the ``knee'' ($\sim 10^{15}$ eV) of the
cosmic-ray spectrum. Bell & Lucek (2001) have suggested that the generated
cosmic rays simultaneously amplify the magnetic field around the supernova
remnant shock to many times its pre-shock value. In this case the acceleration
rate may be significantly increased and protons can easily reach energies up to
$10^{17}$ eV. We use a ``simplified'' box model incorporating the magnetic
field amplification suggested by Bell & Lucek to investigate the resulting
modifications of the cosmic-ray spectrum. The model predicts a spectral break
at high energies, close to the ``knee'' region, and in good accordance with
observations.
