  Given a smooth variety $X$ and an effective Cartier divisor $D \subset X$, we
show that the cohomological Chow group of 0-cycles on the double of $X$ along
$D$ has a canonical decomposition in terms of the Chow group of 0-cycles ${\rm
CH}_0(X)$ and the Chow group of 0-cycles with modulus ${\rm CH}_0(X|D)$ on $X$.
When $X$ is projective, we construct an Albanese variety with modulus and show
that this is the universal regular quotient of ${\rm CH}_0(X|D)$.
  As a consequence of the above decomposition, we prove the Roitman torsion
theorem for the 0-cycles with modulus. We show that ${\rm CH}_0(X|D)$ is
torsion-free and there is an injective cycle class map ${\rm CH}_0(X|D)
\hookrightarrow K_0(X,D)$ if $X$ is affine. For a smooth affine surface $X$,
this is strengthened to show that $K_0(X,D)$ is an extension of ${\rm
CH}_1(X|D)$ by ${\rm CH}_0(X|D)$.
