An integral equation theory is developed for the structure and thermodynamics of melts of random copolymers. The molecules are modeled as flexible chains with a random sequence of two types of sites, with square well potentials between sites. The polymer reference interaction site model (PRISM) integral equation theory is extended to random copolymers and used to calculate the static correlations and spinodal lines. For threadlike chains and with the incompressible random phase approximation (IRPA), the PRISM theory reduces to a replica field theory or Landau-type theory presented earlier by others. In this limit the theory predicts a macrophase separation for values of the monomer correlation strength, lambda, greater than a critical value. For smaller values of lambda a microphase separation is predicted with strong concentration fluctuations at finite wave vectors. With the hard-core interactions included, however, the predictions of the theory with different closure approximations are qualitatively different from each other. With atomic closure approximations the nature of the spinodal diagram is similar to the IRPA, although fluctuations are predicted to promote microphase separation. With molecular closure approximations the theory predicts that no microphase separation occurs in these systems and that there is a macrophase separation for all values of lambda.