We measure the nondissipative supercurrent in a single InAs self-assembled quantum dot (QD) coupled to superconducting leads. The QD occupation is both tuned by a back-gate electrode and lateral side gate. The geometry of the side gate allows tuning of the QD-lead tunnel coupling in a region of constant electron number with an appropriate orbital state. Using the side-gate effect we study the interplay between Kondo correlations and superconducting pairing on the QD, observing a decrease in the supercurrent when the Kondo temperature is reduced below the superconducting energy gap in qualitative agreement with theoretical predictions. © 2010 The American Physical Society.