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An experimental study of the configurational thermodynamics for a series of near-eutectic Pt_80-xCu_xP₂₀bulk metallic glass-forming alloys is reported where 14 <x< 27. The undercooled liquid alloys exhibit very high fragility that increases asxdecreases, resulting in an increasingly sharp glass transition. With decreasingx, the extrapolated Kauzmann temperature of the liquid,T_K, becomes indistinguishable from the conventionally defined glass transition temperature,T_g. Forx< 17, the observed liquid configurational enthalpy vs.Tdisplays a marked discontinuous drop or latent heat at a well-defined freezing temperature,T_gm. The entropy drop for this first-order liquid/glass transition is approximately two-thirds of the entropy of fusion of the crystallized eutectic alloy. BelowT_gm, the configurational entropy of the frozen glass continues to fall rapidly, approaching that of the crystallized eutectic solid in the low T limit. The so-called Kauzmann paradox, with negative liquid entropy (vs. the crystalline state), is averted and the liquid configurational entropy appears to comply with the third law of thermodynamics. Despite their ultrafragile character, the liquids atx= 14 and 16 are bulk glass formers, yielding fully glassy rods up to 2- and 3-mm diameter on water quenching in thin-wall silica tubes. The low Cu content alloys are definitive examples of glasses that exhibit first-order melting.