Past exam of the topology and Groups and Algebraic Topology course of the University of Oxford / 2007 / B / 8 / iii

Yes: for $n\ge 1$, the sphere $S^n$ is connected, closed and oriented, with $H_0=H_n=\mathbb{Z}$ and no intermediate homology. These groups do not require the manifold to be homeomorphic to a sphere, but the sphere provides the requested example.