We measured persistence exponents theta(phi) of Ostwald ripening in two dimensions, as a function of the area fraction phi occupied by coarsening domains. The values of theta(phi) in two systems, succinonitrile and brine, quenched to their liquid-solid coexistence region, compare well with one another, providing compelling evidence for the universality of the one-parameter family of exponents. For small phi, theta(phi) approximately = 0.39phi, as predicted by a model that assumes no correlations between evolving domains. These constitute the first measurements of persistence exponents in the case of phase transitions with a conserved order parameter.