Ice-albedo feedback is a potentially important destabilizing effect for the climate of terrestrial planets. It is based on the positive feedback between decreasing surface temperatures, an increase of snow and ice cover, and an associated increase in planetary albedo, which then further decreases surface temperature. A recent study shows that for M stars, the strength of the ice-albedo feedback is reduced due to the strong spectral dependence of stellar radiation and snow/ice albedos; that is, M stars primarily emit in the near IR, where the snow and ice albedo is low, and less in the visible, where the snow/ice albedo is high. This study investigates the influence of the atmosphere (in terms of surface pressure and atmospheric composition) on this feedback, since an atmosphere was neglected in previous studies. A plane-parallel radiative transfer model was used for the calculation of planetary albedos. We varied CO₂ partial pressures as well as the H₂O, CH₄, and O₃ content in the atmosphere for planets orbiting Sun-like and M type stars. Results suggest that, for planets around M stars, the ice-albedo effect is significantly reduced, compared to planets around Sun-like stars. Including the effects of an atmosphere further suppresses the sensitivity to the ice-albedo effect. Atmospheric key properties such as surface pressure, but also the abundance of radiative trace gases, can considerably change the strength of the ice-albedo feedback. For dense CO₂ atmospheres of the order of a few to tens of bar, atmospheric rather than surface properties begin to dominate the planetary radiation budget. At high CO₂ pressures, the ice-albedo feedback is strongly reduced for planets around M stars. The presence of trace amounts of H₂O and CH₄ in the atmosphere also weakens the ice-albedo effect for both stellar types considered. For planets around Sun-like stars, O₃ could also lead to a very strong decrease of the ice-albedo feedback at high CO₂ pressures.