  We prove Bloch's formula for the Chow group of 0-cycles with modulus on
smooth projective varieties over finite fields. This shows that the K-theoretic
idele class group with modulus due to Kato-Saito and the cycle-theoretic idele
class group with modulus due to Kerz-Saito coincide for such varieties. The
proof relies on new results in ramification theory.
