The Metropolis-Adjusted Langevin Algorithm (MALA), originally introduced to sample exactly the invariant measure of certain stochastic differential equations (SDEs) on infinitely long time intervals, can also be used to approximate pathwise the solution of these SDEs on finite time intervals. However, when applied to an SDE with a nonglobally Lipschitz drift coefficient, the algorithm may not have a spectral gap even when the SDE does. This paper reconciles MALA's lack of a spectral gap with its ergodicity to the invariant measure of the SDE and finite time accuracy. In particular, the paper shows that its convergence to equilibrium happens at an exponential rate up to terms exponentially small in time-step size. This quantification relies on MALA's ability to exactly preserve the SDE's invariant measure and accurately represent the SDE's transition probability on finite time intervals.