The radical ion HCl+, a key intermediate in the chlorine chemistry of the interstellar gas, has been identified for the first time in the interstellar medium with the Herschel Space Observatory's Heterodyne Instrument for the Far-Infrared. The ground-state rotational transition of H35Cl+, 2Π3/2 J = 5/2-3/2, showing Λ-doubling and hyperfine structure, is detected in absorption toward the Galactic star-forming regions W31C (G10.6-0.4) and W49N. The complex interstellar absorption features are modeled by convolving in velocity space the opacity profiles of other molecular tracers toward the same sources with the fine and hyperfine structure of HCl+. This structure is derived from a combined analysis of optical data from the literature and new laboratory measurements of pure rotational transitions, reported in the accompanying Letter by Gupta et al. The models reproduce well the interstellar absorption, and the frequencies inferred from the astronomical observations are in exact agreement with those calculated using spectroscopic constants derived from the laboratory data. The detection of H37Cl+ toward W31C, with a column density consistent with the expected 35Cl/37Cl isotopic ratio, provides additional evidence for the identification. A comparison with the chemically related molecules HCl and H2Cl+ yields an abundance ratio of unity with both species (HCl+ : H2Cl+ : HCl ~ 1). These observations also yield the unexpected result that HCl+ accounts for 3%-5% of the gas-phase chlorine toward W49N and W31C, values several times larger than the maximum fraction (~1%) predicted by chemical models.