  We prove that any one-relator group $G$ is the fundamental group of a compact
Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to
the fundamental group of a compact Riemann surface of genus g > 0 with at most
one orbifold point of order $n \geq 1$. We also classify all groups of
deficiency at least two that are also the fundamental group of some compact
Sasakian manifold.
