  Let $ G $ be a finite non-solvable group with a primitive irreducible
character $ \chi $ that vanishes on one conjugacy class. We show that $ G $ has
a homomorphic image that is either almost simple or a Frobenius group. We also
classify such groups $ G $ with a composition factor isomorphic to a sporadic
group, an alternating group $ \rm{A}_{n} $, $ n\geq 5 $ or $ \rm{PSL}_{2}(q) $,
where $ q\geq 4 $ is a prime power, when $ \chi $ is faithful. Our results
partially answer a question of Dixon and Rahnamai Barghi.
