The paper considers an evolution of the free boundary of a finite volume of a self-gravitating ideal gas moving in the vacuum. Unsteady flows are described by a phenomenological mathematical model, which has the form of a system of nonlinear Volterra integro-differential equations written in Eulerian coordinates. The gas volume moves in a force field generated by the Newtonian potential in general form. The boundary conditions are specified on the free gas-vacuum boundary, which is a priori unknown and determined simultaneously with gas flow construction. Conversion to Lagrangian coordinates allows us to reduce the original problem to an equivalent one, which consists of Volterra integral equations and the continuity equation in Lagrangian form, with Cauchy conditions specified for all these equations. Therefore, the application of Lagrangian coordinates makes it possible, in particular, to eliminate the unknown boundary. The theorem of the existence and uniqueness of the solution in the space of infinitely differentiable functions is proved for this problem. The free boundary is determined as an image of the surface bounding the gas-filled region in reverse transition. Herewith, the method for studying the free boundary is similar to the approach that the authors apply to studying the dynamics of the rarefied mass of a self-gravitating gas. Nu-merical calculations of gas flow are made, including the construction of the free gas-vacuum boundary. The influence of gravitation and the initial gas particle velocity on the formation of gas cloud configuration in the vacuum and on cloud evolution is studied. The results are of interest in terms of solving relevant astrophysical and cosmogonic problems.