A geometrically conservative one-dimensional (1D) arbitrary Lagrangian-Eulerian (ALE) version of the advective upstream splitting method (AUSM) shock capturing scheme is presented. The spatial discretization is based on a modified form of AUSM which splits the flux vector according to the eigenvalues of the compressible Euler system in ALE form and recovers the original flux vector splitting in the absence of grid movement. The generalized form of AUSM is given the name AUSM(ALE). Extension to second-order accuracy is achieved by a piecewise linear reconstruction of the dependent variables with total variation diminishing limiting of slopes. The ALE formulation is completed by incorporating an implicit time-averaged normals form of the geometric conservation law for cylindrically and spherically symmetric time-dependent finite volumes which is valid for any two-level time-integration method. The effectiveness of the method for both fixed and moving grids is demonstrated via several 1D test problems including a standard shock tube problem and an infinite strength reflected shock problem. The method is then applied to a benchmark spherically symmetric underwater explosion problem to demonstrate the efficacy of the numerical procedure for problems of this type. In the two-phase detonation problem the spherical surface separating the expanding detonation-products gas bubble and the surrounding water is explicitly tracked as a Lagrangian surface using AUSM(ALE) in conjunction with appropriate equations of state describing the detonation-products gas and water phases. The basic features of the spherically symmetric detonation problem are discussed such as shock/free-surface interaction and late time hydrodynamics.