  We consider a three-dimensional topological insulator (TI) wire with a
non-uniform chemical potential induced by gating across the cross-section. This
inhomogeneity in chemical potential lifts the degeneracy between two
one-dimensional surface state subbands. A magnetic field applied along the
wire, due to orbital effects, breaks time-reversal symmetry and lifts the
Kramers degeneracy at zero-momentum. If placed in proximity to an $s$-wave
superconductor, the system can be brought into a topological phase at
relatively weak magnetic fields. Majorana bound states (MBSs), localized at the
ends of the TI wire, emerge and are present for an exceptionally large region
of parameter space in realistic systems. Unlike in previous proposals, these
MBSs occur without the requirement of a vortex in the superconducting pairing
potential, which represents a significant simplification for experiments. Our
results open a pathway to the realisation of MBSs in present day TI wire
devices.
