We report the probable detection of the 13CH+ (1-0) line in absorption against the dust continuum emission of a bright star-forming region, G10.6-0.4. If the 13CH+ line velocity centroid is that of the H I gas seen in absorption, the rest frequency we determine is 830.132(3) GHz. This is to be compared to the value of 830.131 GHz derived from new direct spectroscopy laboratory measurements of 12CH+ (1-0) and scaling from 12CH+ (1-0) to 13CH+ (1-0). However, if the 13CH+ line originated in either the star-forming region or in the intervening molecular gas as traced by 12CO (1-0) emission, then the inferred rest frequency would be 830.107(1) or 830.193(4) GHz, respectively, both clearly out of the predicted range. We thus conclude that the 13CH+ line arises in a gas phase not traced by the 12CO (1-0) line but one containing a large column density of cold H I and rich enough in H2 to form CH+. Such properties are those of the cold neutral medium. The inferred column density is N(13CH+)=(2.2+/-0.4)×1012 cm-2, corresponding to N(CH+)=(8.8+/-1.7)×1013 cm-2 for an isotopic abundance ratio [12CH+]/[13CH+]=40. With some assumptions concerning the H2 content of the atomic gas, we estimate the total hydrogen column density in this velocity range and find a CH+ abundance relative to H about twice that in the solar neighborhood, as derived from optical absorption lines.