The stability of polynomials with conic uncertainty is analysed, i.e., a convex cone of directions is known a priori within which the coefficient vector of the nominal polynomial is being perturbed. This corresponds to dropping the requirement in previous work that the convex set be absorbing, i.e., the convex set does not have to contain the origin. Thus, the set of all possible perturbed polynomials is no longer an affine space but a conic set. It is of interest to determine the stability radius of a stable nominal polynomial with respect to perturbations in the direction of an arbitrary nonempty compact convex set. Dropping the assumption that the convex set be absorbing, considerably complicates the theory. In particular, the stability radius of the nominal polynomial may become infinite in certain directions of the convex set. This is the main subject of the paper. Necessary and sufficient conditions for a conic set of polynomials to be Hurwitz stable are derived. The analytical tools derived include an edge theorem and Rantzer-type conditions for marginal stability (semistability). The results are applied to prove an extremal-ray result for conic sets whose cone of directions is given by an interval polynomial.