  We show, for a smooth projective variety $X$ over an algebraically closed
field $k$ with an effective Cartier divisor $D$, that the torsion subgroup
$\CH_0(X|D)\{l\}$ can be described in terms of a relative {\'e}tale cohomology
for any prime $l \neq p = {\rm char}(k)$. This extends a classical result of
Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We
prove the Roitman torsion theorem (including $p$-torsion) for $\CH_0(X|D)$ when
$D$ is reduced. We deduce applications to the problem of invariance of the
prime-to-$p$ torsion in $\CH_0(X|D)$ under an infinitesimal extension of $D$.
