Given a smooth variety X and an effective Cartier divisor D subset of 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 CH0(X) and the Chow group of 0-cycles with modulus CH0(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 CH0(X|D). As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that CH0(X|D) is torsion-free and there is an injective cycle class map CH0(X|D) hooked right arrow 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 CH1(X|D) by CH0(X|D).