In this paper we give a procedure for finding rational solutions of a given first-order ODE with functional and constant coefficients which occur in a rational way. We derive an associated system with the same solvability, and sufficient and necessary conditions for the existence of rational solutions are given. In the case where all parametric coefficients are constant, we give an algorithm to compute the rational solutions. In the case where one functional coefficient appears, we algorithmically find rational general solutions which rationally depend on the appearing transcendental constant. In the other cases, the presented procedure is not completely algorithmic.