Using only the known short-distance behavior of quantum chromodynamics it is possible to prove that, for sufficiently large quark mass m, and fixed antiquark mass, the dimeson (${Q}^{2}$q\ifmmode\bar\else\textasciimacron\fi{} $^{2}$) must be stable against strong decay into two mesons. The binding energy is (1/9m${\ensuremath{\alpha}}_{s}^{2}$[1+O(${m}^{\mathrm{\ensuremath{-}}1}$)]. We then study systems in which Q is c or b, using the many-body confining interaction that comes from a Born-Oppenheimer approximation to the MIT bag model. The calculations were performed using the Green's-function Monte Carlo method. The ${1}^{+}$ isoscalar dimeson T(bb\ifmmode \bar{u}\else \={u}\fi{}d\ifmmode\bar\else\textasciimacron\fi{}) is bound by \ensuremath{\sim}70 MeV with respect to two B mesons; it can only decay weakly, therefore. The calculations of the dimesons (ccq\ifmmode\bar\else\textasciimacron\fi{} q\ifmmode\bar\else\textasciimacron\fi{}') and (bcq\ifmmode\bar\else\textasciimacron\fi{} q\ifmmode\bar\else\textasciimacron\fi{} ') are more uncertain, but indicate that the latter may also be bound.