The thermal conductivity kappa of the heavy-fermion superconductor CeIrIn5 was measured as a function of temperature down to T(c)/8, for current directions parallel (J parallel c) and perpendicular (J parallel a) to the tetragonal c axis. For J parallel a, a sizable residual linear term kappa(0)/T is observed, as previously, which confirms the presence of line nodes in the superconducting gap. For J parallel c, on the other hand, kappa/T-->0 as T-->0. The resulting precipitous decline in the anisotropy ratio kappa(c)/kappa(a) at low temperature rules out a gap structure with line nodes running along the c axis, such as the d-wave state favored for CeCoIn5, and instead points to a hybrid gap of E(g) symmetry.