This paper addresses the problem of designing stabilizing fixed order output feedback controllers for two-dimensional mixed continuous-discrete-time systems. The paper starts by addressing this problem in a basic formulation, where the plant is supposed strictly proper, and the output feedback controller is supposed static. It is shown that a necessary and sufficient condition for establishing the existence of a stabilizing feedback controller can be obtained by solving two convex optimization problems with linear matrix inequality (LMI) constraints. The condition is mainly obtained by exploiting a reformulation of the problem based on the stability of a matrix over the complex unit disc, and by introducing suitable tables for establishing stability of polynomials with complex coefficients. The condition is sufficient for any size of the LMIs, and is also necessary for a size sufficiently large. Then, the paper proceeds by addressing a more general formulation, showing that the proposed approach can be used also when the plant is allowed to be nonstrictly proper, and when the output feedback controller is allowed to be dynamic.