Explicitly correlated basis set of Gaussian-type geminals has been employed in supermolecular calculations of the interaction energy of two helium atoms using the second- and third-order of the many-body perturbation theory and the Moller-Plesset partitioning of the Hamiltonian. A geminal extension of the counterpoise procedure of Boys and Bernardi has been proposed to correct for the basis set superposition error. Performance of the proposed correction scheme has been analyzed at the second-order level using a sequence of geminal bases varying in the degree of completeness in representing the intra- and intermonomer correlation effects. The nonlinear parameters of these bases were optimized by minimizing the second-order energy of the helium atom and the second-order dispersion energy of the He dimer. The best upper bounds to date have been obtained for both quantities. The numerical results show that the counterpoise procedure should be used at all levels of basis set completeness. By employing the union of the largest of the obtained bases and reoptimizing some of the nonlinear parameters using the complete second-order energy functional for the dimer, the best estimates to date of the second- and third-order supermolecular interaction energies for He-2 have been computed. At the minimum interatomic separation these energies are estimated to be accurate to 0.01 K or better. Adding higher-order terms computed using orbital bases, leads to a helium dimer interaction potential with the depth of 11.00 K, somewhat larger than current experimental results.