Water-Nafion phase equilibria and proton conductivities were measured in two ways. First, Nafion was in contact with saturated water vapor. Second, Nafion was in contact with liquid water at the same temperature. At 29 degrees C, for preboiled, vapor-equilibrated Nafion exposed to water with an activity = 1 and air pressures ranging from 0 to 0.96 bar, the water content was lambda = 23 +/- 1 mol H2O/mol SO3-. For the preboiled, liquid-equilibrated membrane, lambda = 24 +/- 2. At 100% relative humidity (RH), the water content of preboiled Nafion decreased as the temperature rose from 30 to 80 degrees C but did not recover its initial water content when the temperature returned to 30 degrees C. The water content of predried Nafion at 1 atm and 30 degrees C was lambda = 13.7 +/- 0.2 when vapor-equilibrated and lambda = 13.1 +/- 0.5 when liquid-equilibrated. A Nafion membrane originally boiled in water had much higher liquid- and 100% RH vapor-equilibrated proton conductivities than the same membrane originally dried at 110 degrees C with a RH less than 2%. The liquid-equilibrated and 100% RH vapor-equilibrated membrane conductivities were the same when the membrane had the same thermal history. The conductivity data was fit to a model, and the water content was determined at different temperatures. The predried membrane water content increased with temperature, and the preboiled membrane's water content changed slightly with temperature. Both water sorption and proton-conductivity data do not exhibit Schroeder's paradox. These studies and previous results suggest that Schroeder's paradox is resolved when attention is given to the thermal history of the absorbing polymer.