Inversion problems have recently drawn vast amounts of attention from the physics community due to their potential widespread applications. In this Rapid Communication, a different type of inversion problem in physics is proposed: an inverse emissivity problem, which aims to determine the emissivity g(nu) by measuring only the total radiated power J(T). Like other inverse problems, this one has potential for important practical applications. An exact solution is obtained for the proposed inverse problem. A unique existence theorem and techniques for eliminating divergences are also presented. A universal function set (UFS) suggested for numerical calculations is shown to be very useful in a numerical example. The UFS makes this inversion method practical and convenient for realistic calculations.