We report measurements of turbulent heat transport in samples of ethane (C2H6) heated from below while the applied temperature difference DeltaT straddled the liquid-vapor coexistence curve T(phi)(P). When the sample top temperature T(t) decreased below T(phi), droplet condensation occurred and the latent heat of vaporization H provided an additional heat-transport mechanism. The effective conductivity lambda(eff) increased linearly with decreasing T(t), and reached a maximum value lambda(eff)(*) that was an order of magnitude larger than the single-phase lambda(eff). As P approached the critical pressure, lambda(eff)(*) increased dramatically even though H vanished. We attribute this phenomenon to an enhanced droplet-nucleation rate as the critical point is approached.