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The lightest neutral Higgs boson in the minimal supersymmetric extension of the standard model has a tree-level mass less than that of the Z0. We calculate radiative corrections to its mass and to that of the heavier CP-even neutral Higgs boson. We find large corrections that increase with the top quark and squark masses, and vary with the ratio of vacuum expectation values ν2/ν1. These radiative corrections can be as large as O(100) GeV, and have the effect of (i) invalidating lower bounds on ν2/ν1 inferred from unsuccessful Higgs searches at LEP I, (ii) in many cases, increasing the mass of the lighter CP-even Higgs boson beyond mz, (iii) often, increasing the mass of the heavier CP-even Higgs boson beyond the LEP reach, into a range more accessible to the LHC or SSC