For the metastable states of antiprotonic helium atoms, {sup 3}He{sup +}p, the fine and hyperfine splittings of energy levels are calculated within the framework of the Breit-Pauli Hamiltonian. The latter represents the {alpha}{sup 2} order relativistic corrections to the nonrelativistic energy. Results include as well the leading {alpha}{sup 3} order contribution due to the anomalous magnetic moment of electron. The final relative accuracy of the hyperfine intervals is estimated to be about 5x10{sup -5}.