  Many studies have been made on extensions of the standard model with $B-L$
gauge symmetry. The addition of three singlet (right-handed) neutrinos renders
it anomaly-free. It has always been assumed that the spontaneous breaking of
$B-L$ is accomplished by a singlet scalar field carrying two units of $B-L$
charge. This results in a very natural implementation of the Majorana seesaw
mechanism for neutrinos. However, there exists in fact another simple
anomaly-free solution which allows Dirac or inverse seesaw neutrino masses. We
show for the first time these new possibilities and discuss an application to
neutrino mixing with $S_3$ flavour symmetry.
