An asymptotic preserving, all-Mach-number, Godunov-type finite volume scheme is presented for the numerical solution of the compressible 
 Euler equations of gas dynamics. It is well known (S. Kleinerman and A. Majda, 1982) that the purely hyperbolic compressible Euler equations converge
 to the mixed hyperbolic-elliptic incompressible Euler equations when the Mach number tends to zero. In this limit, the numerical schemes for the Euler 
 equations suffer from stiffness, loss of accuracy and stability problems. Here, a single time-scale and multiple space-scales asymptotic analysis (R. Klein, 
 1995) is used to split the Euler fluxes into stiff acoustic and non-stiff convective parts. A semi-implicit discretisation leads to a stable scheme which 
 is asymptotic preserving; i.e. it provides a consistent discretisation in both the compressible and the incompressible regimes. In the new scheme the mass 
 and momentum equations are solved explicitly, whereas the  energy equation leads to an elliptic equation for the pressure which complies with the divergence 
 constraint on the velocity. The time-step is solely determined by the non-stiff convective fluxes and it is independent of the Mach number. The results of 
 some benchmark problems are presented, which validate the new scheme. Part of this work is joint  with Sebastian Noelle, Maria Lukacova andClaus-Dieter Munz.