  We combine Hilbert module and algebraic techniques to give necessary and
sufficient conditions for the existence of an Hermitian torsion-free connection
on the bimodule of differential one-forms of a first order differential
calculus. In the presence of the extra structure of a bimodule connection, we
give sufficient conditions for uniqueness. We prove that any
$\theta$-deformation of a compact Riemannian manifold admits a unique Hermitian
torsion-free bimodule connection and provide an explicit construction of it.
Specialising to classical Riemannian manifolds yields a novel construction of
the Levi-Civita connection on the cotangent bundle.
