The competition between chain formation and phase separation in the Stockmayer fluid (SF) of dipolar particles is analyzed using a renormalized Flory-Huggins model of equilibrium polymerization. Calculated critical properties (T(c), phi(c), Z(c)) for the SF compare favorably with simulations over a wide range of the dimensionless dipolar (or "sticking") energy mu*. We find that the polymerization transition preempts phase separation for a large mu*, i.e., (mu*)2 >22.