The diffuse interstellar bands (DIBs), ubiquitous absorption features in astronomical spectra, have been known since early this century and now number more than a hundred. Ranging from 4,400 angstroms to the near infrared, they differ markedly in depth, width and shape, making the concept of a single carrier unlikely. Whether they are due to gas or grains is not settled, but recent results suggest that the DIB carriers are quite separate from the grains that cause visual extinction. Among molecular candidates the polycyclic aromatic hydrocarbons (PAHs) have been proposed as the possible carriers of some of the DIBs, and we present here laboratory measurements of the optical spectrum of the pyrene cation C16H10+ in neon and argon matrices. The strongest absorption feature falls at 4,435 +/- 5 angstroms in the argon matrix and 4,395 +/- 5 angstroms in the neon matrix, both close to the strong 4,430-angstroms DIB. If this or a related pyrene-like species is responsible for this particular band, it must account for 0.2% of all cosmic carbon. The ion also shows an intense but puzzling broad continuum, extending from the ultraviolet to the visible, similar to what is seen in the naphthalene cation and perhaps therefore a common feature of all PAH cations. This may provide an explanation of how PAHs convert a large fraction of interstellar radiation from ultraviolet and visible wavelengths down to the infrared.