  In this paper, a dissipative version of a compressible one velocity
Baer--Nunziato type system for a mixture of two compressible heat conducting
gases is considered. The complete existence proof for weak solutions to this
system was addressed as an open problem in \cite[Section 5]{KNAC}. The purpose
of this paper is to prove the global in time existence of weak solutions to the
one velocity Baer--Nunziato type system for arbitrary large initial data. The
goal is achieved in three steps. Firstly, the given system is transformed into
a new one which possesses the "Navier-Stokes-Fourier" structure. Secondly, the
new system is solved by an adaptation of the Feireisl--Lions approach for
solving the compressible Navier--Stokes equations. Eventually, the existence of
a weak solution to the original one velocity Baer--Nunziato system using the
almost uniqueness property of renormalized solutions to pure transport
equations.
