  Let X be a K3 surface of degree 8 in P^5 with hyperplane section H. We
associate to it another K3 surface M which is a double cover of P^2 ramified on
a sextic curve C. In the generic case when X is smooth and a complete
intersection of three quadrics, there is a natural correspondence between M and
the moduli space M' of rank two vector bundles on X with Chern classes c_1=H
and c_2=4. We build on previous work of Mukai and others, giving conditions and
examples where M' is fine, compact, non-empty; and birational or isomorphic to
M. We also present an explicit calculation of the Fourier-Mukai transform when
X contains a line and has Picard number two.
