  Two compressible immiscible fluids in 1D and in the isentropic approximation
are considered. The first fluid is surrounded and in contact with the second
one. As the Mach number of the first fluid vanishes, we prove the rigorous
convergence for the fully non--linear compressible to incompressible limit of
the coupled dynamics of the two fluids. A key role is played by a suitably
refined wave front tracking algorithm, which yields precise $\mathbf{BV}$,
$\mathbf{L}^1$ and weak* convergence estimates, either uniform or explicitly
dependent on the Mach number.
