Intelligent reflecting surface (IRS) has gained tremendous attention recently as a disruptive technology for beyond 5G networks. In this paper, we consider the problem of secrecy rate maximization for an IRS-assisted Gaussian multiple-input multiple-output multi-antenna-eavesdropper (MIMOME) wiretap channel (WTC). In this context, we aim to jointly optimize the input covariance matrix and the IRS phase shifts to maximize the achievable secrecy rate of the considered system. To solve the formulated problem which is non-convex, we propose an iterative method based on the block successive maximization (BSM), where each iteration is done in closed form. More specifically, we maximize a lower bound on the achievable secrecy rate to update the input covariance matrix for fixed phase shifts, and then maximize the (exact) achievable secrecy rate to update phase shifts for a given input covariance.We consider the total free space path loss (FSPL) in this system to emphasize the first-order measure of the applicability of the IRS in the considered communication system. We present a convergence proof and the associated complexity analysis of the proposed algorithm. Numerical results are provided to demonstrate the superiority of the proposed method compared to a known solution, and also to show the effect of different parameters of interest on the achievable secrecy rate of the IRS-assisted MIMOME WTC.