  We consider the problem of the orbital upper critical magnetic field,
parallel to the most conducting axis of a quasi-one-dimensional layered
superconductor. It is shown that superconductivity can be destroyed through
orbital effects at fields much higher than the so-called Clogston-Chandrasekhar
paramagnetic limiting field, $H_p$, provided that superconducting pairing of
electrons are of a triplet nature. We demonstrate that the superconducting
state of the quasi-one-dimensional layered conductor,
$\mathrm{Li_{0.9}Mo_6O_{17}}$, is well described by the suggested theory. To
this end, we consider two competing scenarios: 1: a superconducting order
parameter without zeros on the Fermi surface, and 2: one with zeros on the
Fermi surface - both are shown to lead to destruction of superconductivity at a
magnetic field, $H^x_{c_2}$, five times higher than $H_p$. With recent
experimental measurements on the $\mathrm{Li_{0.9}Mo_6O_{17}}$ favoring the
nodeless order parameter, we present a strong argument supporting triplet
pairing in this compound.
