  We construct, using harmonic superspace and the quaternionic quotient
approach, a quaternionic-K\"ahler extension of the most general two centres
hyper-K\"ahler metric. It possesses $U(1)\times U(1)$ isometry, contains as
special cases the quaternionic-K\"ahler extensions of the Taub-NUT and
Eguchi-Hanson metrics and exhibits an extra one-parameter freedom which
disappears in the hyper-K\"ahler limit. Some emphasis is put on the relation
between this class of quaternionic-K\"ahler metrics and self-dual Weyl
solutions of the coupled Einstein-Maxwell equations. The relation between our
explicit results and the recent general ansatz of Calderbank and Pedersen for
quaternionic-K\"ahler metrics with $U(1)\times U(1)$ isometries is traced in
detail.
