  The existence and stability of solitons in Bose-Einstein condensates with
attractive inter-atomic interactions, described by the Gross-Pitaevskii
equation with a three-dimensional (3D) periodic potential, are investigated in
a systematic form. We find a one-parameter family of stable 3D solitons in a
certain interval of values of their norm, provided that the strength of the
potential exceeds a threshold value. The minimum number of $^{7}$Li atoms in
the stable solitons is 60, and the energy of the soliton at the stability
threshold is $\approx 6$ recoil energies in the lattice. The respective
energy-vs.-norm diagram features two cuspidal points, resulting in a typical
\textit{swallowtail pattern}, which is a generic feature of 3D solitons
supported by low- (2D) or fully-dimensional lattice potentials.
