PKS 1413+135 (z = 0.24671) is one of very few radio-loud active galactic nuclei (AGNs) with an apparent spiral host galaxy. Previous authors have attributed its nearly exponential infrared cutoff to heavy absorption but have been unable to place tight limits on the absorber or its location in the optical galaxy. In addition, doubts remain about the relationship of the AGN to the optical galaxy given the observed lack of reemitted radiation. We present new Hubble Space Telescope (HST), ASCA, and Very Long Baseline Array observations, which throw significant new light on these issues. The HST observations reveal that the active nucleus of PKS 1413+135 has an extremely red color, (V-H) = 6.9 mag, requiring both a spectral turnover at a few microns because of synchrotron aging and an absorbing region the size of a giant molecular cloud. Combining constraints from the HST and ASCA data, we derive an intrinsic column NH = 4.6 × 1022 cm-2 and covering fraction f = 0.12. The spin temperature of the molecular absorption lines found by previous authors suggests that the cloud is located in the disk of the optical galaxy, making our sight line rather unlikely (P ~ 2 × 10-4). The properties of this region appear typical of large giant molecular clouds in our own Galaxy. The H I absorber appears centered 25 mas away from the nucleus, while the X-ray and nearly all of the molecular absorbers must cover the nucleus, implying a rather complicated geometry and cloud structure, in particular requiring a molecular core along our line of sight to the nucleus. Interestingly, the HST/NICMOS data require the AGN to be decentered relative to the optical galaxy by 13 ± 4 mas. This could be interpreted as suggestive of an AGN location far in the background compared with the optical galaxy, but it can also be explained by obscuration and/or nuclear structure, which is more consistent with the observed lack of multiple images.