The coarsening of a random interface in a fluid of surface tension gamma and viscosity mu is analyzed using a curvature distribution, function A (K-m,,t) which gives the distribution of the mean curvature K-m and Gaussian curvature K-g on the interface. There is a variation in the area distribution function due to the rate of change of K-m, K-g and the compression of the interface due to tangential motion. The rates of change of mean and Gaussian curvature at a point are related to the rate of change of the normal velocity in the tangential directions along the interface. The fluid velocity is governed by the Stokes equation for a viscous flow, and the velocity field at a point is determined as an integral, of the product of the Oseen tensor and the normal force at other points on the interface. Using a general form for this integral, it is shown that there is a characteristic variable K* = K-g/(K-m(2) - 4K(g))(1/2) which isindependent of time even as K-m and K-g decrease proportional to t(-1) and t(-2) respectively. In the late stages, analytical forms for the distribution function are determined m the limit K-m much less than K* using a similarity variable eta=(gamma K(m)t/mu). Two reasonable approximations are used for the characteristic length for the-correlation of the curvature and normal along,the interface, and the results for these two approximations are quadratic polynomials in \eta\ which are nonzero for a finite interval about eta=0. It is expected that the actual distribution function is in between these two limiting cases.