Photoconduction in the monoclinic phase of quasi-one-dimensional conductor TaS$_3$ has been observed at $T < 70$~K. It was studied jointly with low-temperature ohmic and non-linear dark conduction. The strong sample quality dependence of both photoconduction and dark conduction at this temperature region has been observed. Together with a similarity of the main features of the photoconduction characteristic of both monoclinic ({it m-}TaS$_3$) and orthorhombic ({it o-}TaS$_3$) samples the following new peculiarities of photoconduction in {it m-}TaS$_3$ were found: 1) the dependence of the activation energy of photoconduction on temperature, $T$, 2) the change of the recombination mechanism from the linear type to the collisional one at low $T$ with a sample quality growth, 3) the existence of a fine structure of the electric-field dependence of photoconduction. Spectral study gives the Peierls energy gap value $2Delta ^*= 0.18$~eV.
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