We have completed a long-slit, spectrophotometric survey of the optical continuum emission of 20 planetary nebulae (PNs) covering a wide range of C/O ratios and have detected the broad, red emission band known as extended red emission (ERE) in seven of these objects. We have therefore added PNs to the growing list of astronomical objects which are known to emit ERE. In PNs where ERE is detected, the band is found to have an average integrated intensity of 1.9 x 10(-3) ergs cm2 s-1 sr-1 and to contribute approximately 15% of the flux in the continuum between 550 and 820 nm. The ERE is detected as emission in excess of a theoretical atomic continuum, computed as described by Brown & Mathews, and scaled to match the observed nebular spectrum in the 320-400 nm spectral region. The input parameters necessary to compute such a model for a given PN are the electron temperature, T(e), the color excess, E(B - V), and the helium abundance ratios, He+/H+ and He++/H+; these were compiled, along with C/O ratio measurements, from an extensive literature search. We compare the ERE band observed in planetary nebulae to the ERE band observed in reflection nebulae and find they are similar in band profile and intensity. We find that the presence or absence of ERE in PNs is correlated with the C/O ratio in the sense that none of the oxygen-rich nebulae in our sample exhibit the ERE band above our detection limit, while most carbon-rich nebulae do. We claim that this supports the identification of ERE as the photoluminescence of hydrogenated amorphous dust grains. Our data also suggest that ERE and the so-called unidentified infrared emission bands (UIB) are not necessarily produced by the same carrier because we present examples of several PNs where the UIB's are detected and ERE is not.