The instability of an earlier perfectly matched layer (PML) formulation for the linearized Euler equations is investigated. It is found that, in the presence of a mean flow, there exist acoustic waves that have a positive group velocity but a negative phase velocity in the direction of the mean flow and these waves become actually amplified in the previous formulation, thus giving rise to the instability. A new stable PML formulation that is perfectly matched to the Euler equations and does not entail exponentially growing solution is presented. Furthermore, the new formulation is given in unsplit physical variables which should facilitate its implementation in many practical schemes. In addition, the well-posedness of the new formulation is also considered. It is shown that the proposed equations are well-posed for horizontal y-layers but weakly well-posed for vertical x-layers and corner layers. However, it is further shown that they can be easily modified to be symmetrizable, thus strongly well-posed, by an addition of arbitrarily small terms. Numerical examples that verify the stability and effectiveness of the proposed PML equations, such as an absorbing boundary condition, are given.