This paper considers the problem of stabilizing a given linear time-invariant plant using a fixed order compensator. As a first step, attention is focussed on the constant gain stabilization problem. An appropriately generalized version of the Hermite-Biehler Theorem is given and shown to be useful in providing a solution to this problem. A complete analytical characterization of all stabilizing feedback gains is provided as a closed form solution. This is in stark contrast to the highly nonlinear inequalities that one may have to deal with in trying to solve this problem using the Routh-Hurwitz criterion or the graphical procedures which must be followed while using the Nyquist criterion or the Root Locus technique. The development of similar approaches for stabilization using PID or first order controllers is a topic of current research.