  We show that the one-loop quantum deformation of the universal hypermultiplet
provides a family of complete $1/4$-pinched negatively curved quaternionic
K\"ahler (i.e. half conformally flat Einstein) metrics $g^c$, $c\ge 0$, on
$\mathbb R^4$. The metric $g^0$ is the complex hyperbolic metric whereas the
family $(g^c)_{c>0}$ is equivalent to a family of metrics $(h^b)_{b>0}$
depending on $b=1/c$ and smoothly extending to $b=0$ for which $h^0$ is the
real hyperbolic metric. In this sense the one-loop deformation interpolates
between the real and the complex hyperbolic metrics. We also determine the
(singular) conformal structure at infinity for the above families.
