  Let $R$ be a regular semi-local ring, essentially of finite type over an
infinite perfect field of characteristic $p \ge 3$. We show that the cycle
class map with modulus from an earlier work of the authors induces a
pro-isomorphism between the additive higher Chow groups of relative 0-cycles
and the relative $K$-theory of truncated polynomial rings over $R$. This
settles the problem of equating 0-cycles with modulus and relative $K$-theory
of such rings via the cycle class map.
