  Recent results from lattice QCD simulations provide a realistic picture,
based upon first principles, of~$\Upsilon$ physics. We combine these results
with the experimentally measured mass of the $\Upsilon$~meson to obtain an
accurate and reliable value for the $b$-quark's pole mass. We use two different
methods, each of which yields a mass consistent with $M_b = 5.0(2)$~GeV. This
corresponds to a bare mass of $M_b^0 = 4.0(1)$~GeV in our lattice theory and an
$\msbar$~mass of $M_b^\msbar(M_b)=4.0(1)$~GeV. We discuss the implications of
this result for the $c$-quark mass.
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THIS IS THE VERSION WHICH WILL BE PUBLISHED IN PRL. SUBSTANTIAL MATERIAL HAS
BEEN ADDED, INCLUDING RESULTS WITH DYNAMICAL FERMIONS AND A CALCULATION OF THE
MSBAR MASS.
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