  We show that the $\nu=8$ integer quantum Hall state can support Majorana zero
modes at domain walls between its two different stable chiral edge phases
without superconductivity. This is due to the existence of an edge phase that
does not support gapless fermionic excitations; all gapless excitations are
bosonic in this edge phase. Majorana fermion zero modes occur at a domain wall
between this edge phase and the more conventional one that does support gapless
fermions. Remarkably, due to the chirality of the system, the topological
degeneracy of these zero modes has exponential protection, as a function of the
relevant length scales, in spite of the presence of gapless excitations,
including gapless fermions. These results are compatible with charge
conservation, but do not require it. We discuss generalizations to other
integer and fractional quantum Hall states, and classify possible mechanisms
for appearance of Majorana zero modes at domain walls.
