This note is concerned with the accuracy of the solution of nearly uncoupled Markov chains by a direct method based on the LU decomposition. It is shown that plain Gaussian elimination may fail in the presence of rounding errors. A modification of Gaussian elimination with diagonal pivoting as well as corrections of small pivots by sums of off-diagonal elements in the pivoting columns is proposed and analyzed. It is shown that the accuracy of the solution is affected by two condition numbers associate with the aggregate and the coupling respectively. (Also cross-referenced as UMIACS-TR-90-95) Appeared in Numerische Mathematik 59 (1991) 1--12.