Interstellar dust (ISD) is the condensed phase of the interstellar medium. In situ data from the Cosmic Dust Analyzer on board the Cassini spacecraft reveal that the Saturnian system is passed by ISD grains from our immediate interstellar neighborhood, the local interstellar cloud. We determine the mass distribution of 36 interstellar grains, their elemental composition, and a lower limit for the ISD flux at Saturn. Mass spectra and grain dynamics suggest the presence of magnesium-rich grains of silicate and oxide composition, partly with iron inclusions. Major rock-forming elements (magnesium, silicon, iron, and calcium) are present in cosmic abundances, with only small grain-to-grain variations, but sulfur and carbon are depleted. The ISD grains in the solar neighborhood appear to be homogenized, likely by repeated processing in the interstellar medium.