  The Hopf algebra dual form for the non--standard uniparametric deformation of
the (1+1) Poincar\'e algebra $iso(1,1)$ is deduced. In this framework, the
quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite
dimensional Lie algebra. A change in the basis of the dual form is obtained in
order to compare this deformation to the standard one. Finally, a non--standard
quantum Heisenberg group acting on a quantum Galilean plane is obtained.
