In three-dimensional topological insulators (3D TI) nanowires, transport occurs via gapless surface states where the spin is fixed perpendicular to the momentum[1-6]. Carriers encircling the surface thus acquire a pi Berry phase, which is predicted to open up a gap in the lowest-energy 1D surface subband. Inserting a magnetic flux ({Phi}) of h/2e through the nanowire should cancel the Berry phase and restore the gapless 1D mode[7-8]. However, this signature has been missing in transport experiments reported to date[9-11]. Here, we report measurements of mechanically-exfoliated 3D TI nanowires which exhibit Aharonov-Bohm oscillations consistent with topological surface transport. The use of low-doped, quasi-ballistic devices allows us to observe a minimum conductance at {Phi} = 0 and a maximum conductance reaching e^2/h at {Phi} = h/2e near the lowest subband (i.e. the Dirac point), as well as the carrier density dependence of the transport.