  The hydrodynamics of an ultrarelativistic flow, enclosed by a strong shock
wave, are described by the well known Blandford-McKee solutions in spherical
geometry. These solutions, however, become inaccurate at a distance $\sim R/2$
behind the shock wave, where $R$ is the shock radius, as the flow approaches
Newtonian velocities. In this work we find a new self-similar solution which is
an extension to the Blandford-McKee solutions, and which describes the interior
part of the blast wave, where the flow reaches mildly relativistic to Newtonian
velocities. We find that the velocity profile of the internal part of the flow
does not depend on the value of the shock Lorentz factor, $\Gamma$, and is
accurate from $r=0$ down to a distance of $R/\Gamma^2$ behind the shock.
Despite the fact that the shock wave is in causal contact with the entire flow
behind it, a singular point appears in the equations. Nevertheless, the
solution is not required to pass through the singular point: for ambient
density that decreases slowly enough, $\rho \propto r^{-k}$ with
$k<\frac{1}{2}(5-\sqrt{10})\cong0.92$, a secondary shock wave forms with an
inflow towards the origin.
