The condensation coefficient of water vapor on liquid water and the thermal accommodation coefficient of air on liquid water are poorly known despite their importance in many applications, such as cloud physics. We have developed a new technique for determining the condensation and thermal accommodation coefficients experimentally. The technique consists of simultaneously measuring the homogeneous nucleation rate of ice and the evaporation rate of liquid water droplets as a function of pressure (droplet Knudsen number). As the Knudsen number increases, surface kinetic processes limit mass and energy fluxes and, as a result, the equilibrium temperature of an evaporating droplet is a function of the condensation and thermal accommodation coefficients. The homogeneous freezing nucleation rate is used as a sensitive measure of the droplet temperature. The nucleation and evaporation rates are determined by observing the scattered light from evaporating water droplets suspended in an electrodynamic levitation system housed within a controlled environment. The observed rates are consistent with a condensation coefficient between 0.04 and 0.1, with 0.06 being most probable, and a thermal accommodation coefficient between 0.1 and 1, with 0.7 being most probable.