The Kelvin-Helmholtz ( KH ) and tearing instabilities are likely to be important for the process of fast magnetic reconnection that is believed to explain the observed explosive energy release in solar flares. Theoretical studies of the instabilities, however, typically invoke simplified initial magnetic and velocity fields that are not solutions of the governing magnetohydrodynamic ( MHD) equations. In the present study, the stability of a reconnecting current sheet is examined using a class of exact global MHD solutions for steady state incompressible magnetic reconnection, discovered by Craig & Henton. Numerical simulation indicates that the outflow solutions where the current sheet is formed by strong shearing flows are subject to the KH instability. The inflow solutions where the current sheet is formed by a fast and weakly sheared inflow are shown to be tearing unstable. Although the observed instability of the solutions can be interpreted qualitatively by applying standard linear results for the KH and tearing instabilities, the magnetic field and plasma flow, specified by the Craig-Henton solution, lead to the stabilization of the current sheet in some cases. The sensitivity of the instability growth rate to the global geometry of magnetic reconnection may help in solving the trigger problem in solar flare research.