We determine the bottom <(MS)over bar> quark mass (m) over bar(b) and the quark mass in the potential subtraction scheme from moments of the <b(b)over bar> production cross section and from the mass of the Upsilon 1S state at next-to-next-to-leading order in a reorganized perturbative expansion that sums Coulomb exchange to all orders. We find (m) over bar(b)((m) over bar(b)) = (4.25 +/- 0.08) GeV and m(b,PS)(2 GeV) = (4.59 +/- 0.08) GeV for the potential-subtracted mass at the scale 2 GeV, adopting a conservative error estimate. (C) 1999 Elsevier Science B.V. All rights reserved.