We study the specific heat of single-crystal UPta for uniaxial stress 0 < 5* < 4 kbar. For Sllc, the lower and upper superconducting transitions move towards each other in temperature and merge by S^2 kbar and T-485 mK. In the normal state, the Sommerfeld constant y increases 12% by 2 kbar and then plateaus. By contrast, for Slla, / decreases steadily and the onset Tf's of the two superconducting transitions remain essentially unchanged, although the specific-heat peaks broaden markedly. Residual strain in the basal plane may account for some crystals of UPt3 showing only one broad superconducting transition. PACS numbers: 74.70.Tx, 74.30.Ek The investigation of anisotropic responses in the heavy-fermion superconductors has revealed essential aspects of the superconducting state. Axial anisotropy in the temperature dependences of the ultrasonic attenuation II] and the magnetic penetration depth [2] has established the nodal structure of the superconducting gap in UPt3. Group-theory considerations [3] then limit the available unconventional superconductor representations, but cannot establish uniquely the nature of the pairing. Pronounced anisotropy in the upper critical field [4] provides further clues, and recent detailed analyses [5,6] of the HciiT) directional dependence allow further constraints on the parity and the order of the non-^-wave pairing mechanism.The discovery of two zero-field superconducting transitions [7] in UPta has opened up a new chapter in heavyfermion physics. In contrast to the marked axial anisotropy of the normal-state-superconductor transition, the double-transition //-T phase diagrams [8,9] are topologically isotropic with respect to field direction. This striking isotropy in an anisotropic superconductor has spurred additional representation scenarios [10] for the superconducting state in UPta and has raised as well concerns about sample homogeneity.In addition to magnetic field, uniaxial stress can serve as a symmetry-breaking field. We report here measurements of the specific heat of single crystals of UPt3 stressed parallel to and perpendicular to the hexagonal basal plane. Our primary result is the first evidence of an anisotropic phase diagram for the superconducting double transition. Moreover, our findings indicate that residual stresses in the basal plane may explain the current mystery of why some UPt3 crystals have only one superconducting peak in the specific heat.Single crystals of UPtj were grown by the verticalfloat-zone refining method, annealed at 950 °C for 12 h, and then slowly cooled. Characterization via ac magnetic susceptibility gives a single 7^=545 mK with a 10%-90% transition width of 7 mK. These samples are prepared similarly to those used in the ultrasonic velocity determination of the double-transition H-T phase diagram [9], samples whose heat capacity has not been measured previously. Uniaxial stress was applied to crystals of typical dimensions 1.5x1.0x1.0 mm^ with 1.5 mm^ parallel faces cut perpendicular to either the a or the c axis. The stress cell ...