  Quaternion Kahler manifolds are known to be the target spaces for matter
hypermultiplets coupled to N=2 supergravity. It is also known that there is a
one-to-one correspondence between 4n-dimensional quaternion Kahler manifolds
and those 4(n+1)-dimensional hyperkahler spaces which are the target spaces for
rigid superconformal hypermultiplets (such spaces are called hyperkahler
cones). In this paper we present a projective-superspace construction to
generate a hyperkahler cone M^{4(n+1)}_H of dimension 4(n+1) from a
2n-dimensional real analytic Kahler-Hodge manifold M^{2n}_K. The latter emerges
as a maximal Kahler submanifold of the 4n-dimensional quaternion Kahler space
M^{4n}_Q such that its Swann bundle coincides with M^{4(n+1)}_H. Our approach
should be useful for the explicit construction of new quaternion Kahler
metrics. The results obtained are also of interest, e.g., in the context of
supergravity reduction N=2 --> N=1, or alternatively from the point of view of
embedding N=1 matter-coupled supergravity into an N=2 theory.
