This report describe the properties of the cation of the PAH naphthalene (C10H8+) isolated in inert gas matrices under conditions relevant to astrophysical environments. As with other PAH cations, C10H8+ absorbs in the visible. C10H8+ discrete absorption bands fall at 6741, 6520, 6376, 6314, 6151, 5965, 5848, 5784, 5658, 5503, 5341, 5114, and 4560 angstrom. The band at 6741 angstrom is the strongest and falls close to the weak 6742 angstrom DIB. Five other weaker bands (6376, 6314, 5848, 5784, and 5503) also fall remarkably close to the positions of known diffuse interstellar bands (DIBs). The possibility that C10H8+ is responsible for some of the diffuse interstellar bands can be tested by searching for new DIBS at 6520, 6151 and 5965 angstrom, other strong naphthalene cation band positions. If C10H8+ is indeed responsible for the 6742 angstrom feature, it accounts for 0.3% of the cosmic carbon. We also report a remarkably intense, but puzzling, very broad continuum extending from the UV to the visible (2000-3800 angstrom, with a weak tail extending to almost-equal-to 5100 angstrom) which seems to be associated with the ion. The molar absorption coefficient (epsilon) at the peak of the continuum is 2.0 x 10(6) dm3 mol-1 cm-1 (sigma(e) = 7.6 x 10(-15) cm2 molecule-1). If a continuum is a general property of PAH cations, this characteristic will have a strong impact on our understanding of how PAHs convert interstellar UV and visible radiation into IR radiation. We also note that neutral naphthalene may contribute to the 2200 angstrom bump. To produce a detectable feature at 2116 angstrom however, requires a high abundance.