  The existence of full heavy dibaryons $\Omega_{ccc}\Omega_{bbb}$,
$\Omega_{ccc}\Omega_{ccc}$ and $\Omega_{bbb}\Omega_{bbb}$ with $J=0,~1,~2,~3$
and $P=\pm1$ are investigated in the framework of a constituent quark model.
The dibaryon composed of six $c$ or $b$ quarks with $J^{P}=0^{+}$ is possible
to be bound, while the dibaryon with the $cccbbb$ configuration is difficult to
form any bound state. We also study the interaction between two full heavy
baryons, and the effective potentials, suggesting the existence of dibaryons
$\Omega_{ccc}\Omega_{ccc}$ and $\Omega_{bbb}\Omega_{bbb}$, and the absence of
binding for $\Omega_{ccc}\Omega_{bbb}$ system. Besides, the calculation of the
low-energy scattering phase shifts of the $\Omega_{ccc}\Omega_{ccc}$ and
$\Omega_{bbb}\Omega_{bbb}$, as well as the scattering length also confirm the
existence of stable full heavy dibaryons $\Omega_{ccc}\Omega_{ccc}$ and
$\Omega_{bbb}\Omega_{bbb}$.
