We have measured resonance strengths and energies for dielectronic recombination (DR) of Fe XIX forming Fe XVIII via N = 2 → N' = 2 and N = 2 → N' = 3 core excitations. All measurements were carried out using the heavy-ion Test Storage Ring at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. We have also calculated these resonance strengths and energies using two independent, state-of-the-art techniques: the perturbative multiconfiguration Breit-Pauli (MCBP) and multiconfiguration Dirac-Fock (MCDF) methods. Overall, reasonable agreement is found between our experimental results and theoretical calculations. The most notable discrepancies are for the 3l3l' resonances. The calculated MCBP and MCDF resonance strengths for the n = 3 complex lie, respectively, ≈47% and ≈31% above the measured values. These discrepancies are larger than the estimated 20% total experimental uncertainty in our measurements. We have used our measured 2 → 2 and 2 → 3 results to produce a Maxwellian-averaged rate coefficient for DR of Fe XIX. Our experimentally derived rate coefficient is estimated to be good to better than ≈20% for k_BT_e ≥ 1 eV. Fe XIX is predicted to form in photoionized and collisionally ionized cosmic plasmas at k_BT_e 1 eV. Hence, our rate coefficient is suitable for use in ionization balance calculations of these plasmas. Previously published theoretical DR rate coefficients are in poor agreement with our experimental results. None of these published calculations reliably reproduce the magnitude or temperature dependence of the experimentally derived rate coefficient. Our MCBP and MCDF results agree with our experimental rate coefficient to within ≈20%.