Two-fluid flowing equilibrium configurations of a helicity-driven spherical torus (HD-ST) in the realistic region including the spherical flux conserver and the coaxial helicity source are numerically determined by using the combination of the finite difference and the boundary element methods. It is found from the numerical results that the ultra low-$q$ HD-ST with diamagnetic toroidal field and high-$\beta$ (volume average $\beta$ value, $\langle\beta\rangle\cong 18\%$) can be obtained by reversing the toroidal field over the whole region and by increasing the toroidal ion flow up to the Alfv${\rm \acute e}$n Mach number $M_{\rm A}\cong 0.7$. The ion diamagnetic drift velocity is larger than the ${\bf E}\times {\bf B}$ drift velocity and has the same direction.