  Given a tame differential calculus over a noncommutative algebra
$\mathcal{A}$ and an $\mathcal{A}$-bilinear pseudo-Riemannian metric $g_0,$
consider the conformal deformation $ g = k. g_0, $ $k$ being an invertible
element of $\mathcal{A}.$We prove that there exists a unique connection
$\nabla$ on the bimodule of one-forms of the differential calculus which is
torsionless and compatible with $g.$ We derive a concrete formula connecting
$\nabla$ and the Levi-Civita connection for the pseudo-Riemannian metric $g_0.$
As an application, we compute the Ricci and scalar curvature for a general
conformal perturbation of the canonical metric on the noncommutative $2$-torus
as well as for a natural metric on the quantum Heisenberg manifold. For the
latter, the scalar curvature turns out to be a negative constant.
