We report on magnetoconductance experiments in ballistic multiwalled carbon nanotubes threaded by magnetic fields as large as 55 T. In the high temperature regime (100 K), giant modulations of the conductance, mediated by the Fermi level location, are unveiled. The experimental data are consistently analyzed in terms of the field-dependent density of states of the external shell that modulates the injection properties at the electrode-nanotube interface, and the resulting linear conductance. This is the first unambiguous experimental evidence of Aharonov-Bohm effect in clean multiwalled carbon nanotubes.