Material ejected during (or immediately following) the merger of two neutron stars may assemble into heavy elements by the r-process. The subsequent radioactive decay of the nuclei can power electromagnetic emission similar to, but significantly dimmer than, an ordinary supernova. Identifying such events is an important goal of future transient surveys, offering new perspectives on the origin of r-process nuclei and the astrophysical sources of gravitational waves. Predictions of the transient light curves and spectra, however, have suffered from the uncertain optical properties of heavy ions. Here we consider the opacity of expanding r-process material and argue that it is dominated by line transitions from those ions with the most complex valence electron structure, namely the lanthanides. For a few representative ions, we run atomic structure models to calculate radiative data for tens of millions of lines. We find that the resulting r-process opacities are orders of magnitude larger than that of ordinary (e.g., iron-rich) supernova ejecta. Radiative transport calculations using these new opacities indicate that the transient emission should be dimmer and redder than previously thought. The spectra appear pseudo-blackbody, with broad absorption features, and peak in the infrared (~1 micron). We discuss uncertainties in the opacities and attempt to quantify their impact on the spectral predictions. The results have important implications for observational strategies to find and study the radioactively powered electromagnetic counterparts to compact object mergers.