  We present quenched results for B meson decay constants using NRQCD b quarks
and O(a) tadpole improved clover light quarks. For the first time, one-loop
matching factors between lattice and continuum currents are incorporated
through O(\alpha/M) taking operator mixing fully into account. This includes an
important O(\alpha a) discretization correction to the heavy-light axial vector
current. We find f_B = 0.147(11)(^{+8}_{-12})(9)(6) MeV and f_{B_s}/f_B =
1.20(4)(^{+4}_{-0}).
  PACS numbers: 12.38.Gc, 12.39.Hg, 13.20.He, 14.40.Nd
