We establish two inequalities of Stein-Weiss type for the Riesz potential operator Iα,γ (B−Riesz potential operator) generated by the Laplace-Bessel differential operator ΔB in the weighted Lebesgue spaces L p,|x|β,γ . We obtain necessary and sufficient conditions on the pa- rameters for the boundedness of Iα,γ from the spaces L p,|x|β,γ to L q,|x|−λ,γ , and from the spaces L 1,|x|β,γ to the weak spaces WL q,|x|−λ,γ . In the limiting case p=Q/α we prove that the modified B−Riesz potential operator �α,γ is bounded from the spaces Lp,|x|β,γ to the weighted B −BMO spaces BMO |x|−λ ,γ . As applications, we get the boundedness of Iα,γ from the weighted B-Besov spaces Bsθ,|x|β,γ to the spaces B sθ,|x|−λ,γ . Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue L p,|x|β,γ and weighted B-Besov spaces B s pθ,|x|β,γ by using the fundamental solution of the B-elliptic equation Δ α/2 B.