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We consider a continuum model describing steady flows of a miscible mixture of two fluids. The densities ${\rho }_i$ of the fluids and their velocity fields $u^{\left(i\right)}$ are prescribed at infinity: ${\rho }_i{|}_{\infty }={\rho }_{i\infty }>0$, $u^{\left(i\right)}{|}_{\infty }=0$. Neglecting the convective terms, we have proved earlier that weak solutions to such a reduced system exist. Here we establish a uniqueness type result: in the absence of the external forces and interaction terms, there is only one such solution, namely ${\rho }_i\equiv {\rho }_{i\infty }$, $u^{\left(i\right)}\equiv 0$, $i=1,2$.