The ground state phase diagram of the half-filled one-dimensional Holstein-Hubbard model contains a charge-density-wave (CDW) phase, driven by the electron-phonon (e-ph) coupling, and a spin-density-wave (SDW) phase, driven by the on-site electron-electron (e-e) repulsion. Recently, the existence of a third phase, which is metallic and lies in a finite region of parameter space between these two gapped phases, has been claimed. We study this claim using a renormalization-group method for interacting electrons that has been extended to include also e-ph couplings. Our method treats e-e and e-ph interactions on an equal footing and takes retardation effects fully into account. We find a direct transition between the spin- and charge-density wave states. We study the effects of retardation, which are particularly important near the transition, and find that Umklapp processes at finite frequencies drive the CDW instability close to the transition. We also perform determinantal quantum Monte Carlo calculations of correlation functions to confirm our results for the phase diagram.