A Dirac equation for the $qbar{q}$ system is formulated in the presence of the confining potential derived from a static solution of the SU(2) Yang-Mills equations. The resulting radial equations define an eigenvalue problem with singular behavior at $r=0$ and $r=ho$, leading to a discrete spectrum of confined states.The system reduces to four sets of coupled radial equations for each of quark and antiquark components, which do not appear to admit closed-form solutions in terms of standard special functions. An approximate treatment is therefore developed to determine the corresponding energy spectrum. The calculated bound-state energies are found to be largely insensitive to the current quark masses and are of the same order as the observed meson masses, suggesting that the dominant contribution to the meson masses originates from the confining dynamics described by the present Yang-Mills solution. The quark and antiquark spectra exhibit an asymmetry associated with the gauge orientation of the underlying monopole-like configuration, although the individual energy eigenvalues are not gauge invariant observables.