The Cheshire Cat is a relatively poor group of galaxies dominated by two luminous elliptical galaxies surrounded by at least four arcs from gravitationally lensed background galaxies that give the system a humorous appearance. Our combined optical/X-ray study of this system reveals that it is experiencing a line of sight merger between two groups with a roughly equal mass ratio with a relative velocity of ~1350 km/s. One group was most likely a low-mass fossil group, while the other group would have almost fit the classical definition of a fossil group. The collision manifests itself in a bimodal galaxy velocity distribution, an elevated central X-ray temperature and luminosity indicative of a shock, and gravitational arc centers that do not coincide with either large elliptical galaxy. One of the luminous elliptical galaxies has a double nucleus embedded off-center in the stellar halo. The luminous ellipticals should merge in less than a Gyr, after which observers will see a massive 1.2-1.5x10^14^ M_{sun}_ fossil group with an M_r_=-24.0 brightest group galaxy at its center. Thus, the Cheshire Cat offers us the first opportunity to study a fossil group progenitor. We discuss the limitations of the classical definition of a fossil group in terms of magnitude gaps between the member galaxies. We also suggest that if the merging of fossil (or near-fossil) groups is a common avenue for creating present-day fossil groups, the time lag between the final galactic merging of the system and the onset of cooling in the shock-heated core could account for the observed lack of well-developed cool cores in some fossil groups.