Compressible multi-fluid flows are notoriously challenging problems for numerical simulation. Fully Eulerian simulations are usually too diffusive to be able to accurately capture contact discontinuities, except with the use of anti-diffusive schemes Desprès et al. on the phase equation. Lagrangian schemes suffer from large mesh deformation which involves costly remeshing. A promising approach is the use of immersed boundary methods to track the position of the contact discontinuity in an Eulerian framework. Among those methods, cut-cells methods have the desirable property of conserving physical quantities (mass, momentum, energy). However, previous works have used cut-cells in combination with level-set methods which do not guarantee such conservation properties \cite{HuKhoo}. In the present work, we extend the cut-cell method in Puscas et al. to the simulation of compressible multi-fluid flows. The main idea is to follow the material interface with a Lagrangian discretization and to apply the conservative cut-cell methodology in both Eulerian fluid phases. A key point is the use of the solution of a two-phase one-dimensional exact Riemann problem at the interface as in Hu et al., the origin of which will be explained in detail. We will present numerical results evaluating the method on classical tests from the literature and verifying the exact conservation properties of the scheme.