The Aharonov-Bohm (AB) phase is known to be acquired when an electron-path encircles a vector potential-source and, equally, when the latter encircles the former. When the source encircles only part of the electronic cloud (and does so adiabatically), it has been claimed in the literature and recently re-asserted by S. Popescu that the adiabatic theorem forbids the acquisition of an AB phase by (the encircled) part of the electron. We show that, contrary to the above prevalent view, phase acquisition in part of an electronic state can occur and that the initial and after-return adiabatic wave functions are, remarkably, co-degenerate solutions of the Hamiltonian.  This degeneracy repeats itself for further, repeated cyclic motion of the source. A new "weakened AB effect" is demonstrated for a situation in which there is no magnetic field and only a weak, ineffective electric field, but nevertheless AB phases are generated.