The one-dimensional theory of the stabilization of collisional drift waves by shear is discussed. The radial eigenfunction problem is solved by asymptotically matching the solution in the region in which the electrons behave isothermally to the solution in the region in which the electron behavior is adiabatic. The resulting dispersion equation is solved analytically in the limits of strong and weak collisionality and it is found that the stability of drift waves depends critically on the precise form of the collision operator in the adiabatic region. With the collision operator in the Landau form it is found that the effect of collisions is to further increase the damping due to the shear so that the so-called collisional drift instability is in reality a highly stable mode. This result is confirmed by a numerical solution of the full eigenvalue problem and the damping decrement is given as a function of collisionality for a range of shear strengths from 10 to 100.