  This paper proposes a new geometric construction of Enriques surfaces. Its
starting point are K3 surfaces with jacobian elliptic fibration which arise
from rational elliptic surfaces by a quadratic base change. The Enriques
surfaces obtained in this way are characterised by elliptic fibrations with a
rational curve as bisection which splits into two sections on the covering K3
surface. The construction has applications to the study of Enriques surfaces
with specific automorphisms. It also allows us to answer a question of
Beauville about Enriques surfaces whose Brauer groups show an exceptional
behaviour. In a forthcoming paper, we will study arithmetic consequences of our
construction.
