The pD and p 4 He total annihilation cross sections have been measured with the Obelix apparatus at LEAR at, respectively, three and two values of the p incident momentum between 70 MeV/c and 36 MeV/c. The values of the pD annihilation cross section, at such low energies, are in agreement with a surprising result obtained in a recent low statistics measurement of the antiprotonic deuterium K α X-rays [M. Augsburger et al., Phys. Lett. B 461 (1999) 417] (see following article), concerning the unexpected narrow width of the antiprotonic deuterium 1s level.