  Cantilever magnetometry has been used to measure the upper critical magnetic
field $H_{c2}$ of the quasi-one dimensional molecular organic superconductor
(TMTSF)$_2$ClO$_4$. From simultaneous resistivity and torque magnetization
experiments conducted under precise field alignment, $H_{c2}$ at low
temperature is shown to reach 5T, nearly twice the Pauli paramagnetic limit
imposed on spin singlet superconductors. These results constitute the first
thermodynamic evidence for a large $H_{c2}$ in this system and provide support
for spin triplet pairing in this unconventional superconductor
