We use solar occultations observed by the Visual and Infrared Mapping Spectrometer on board the Cassini Spacecraft to extract the 1-5 μm transmission spectrum of Saturn, as if it were a transiting exoplanet. We detect absorption from methane, ethane, acetylene, aliphatic hydrocarbons, and possibly carbon monoxide, with peak-to-peak features of up to 90 parts-per-million despite the presence of ammonia clouds. We also find that atmospheric refraction, as opposed to clouds or haze, determines the minimum altitude that could be probed during mid-transit. Self-consistent exoplanet atmosphere models show good agreement with Saturn's transmission spectrum but fail to reproduce a large absorption feature near 3.4 μm, likely caused by gaseous ethane and a C-H stretching mode of an unknown aliphatic hydrocarbon. This large feature is located in one of the Spitzer Space Telescope bandpasses and could alter interpretations of transmission spectra if not properly modeled. The large signal in Saturn's transmission spectrum suggests that transmission spectroscopy of cold, long-period gaseous exoplanets should be possible with current and future observatories. Motivated by these results, we briefly consider the feasibility of using a survey to search for and characterize cold exoplanets that are analogous to Jupiter and Saturn utilizing a target-of-opportunity approach.