  We demonstrate the operation of an atom interferometer based on a weakly
interacting Bose-Einstein condensate. We strongly reduce the interaction
induced decoherence that usually limits interferometers based on trapped
condensates by tuning the s-wave scattering length almost to zero via a
magnetic Feshbach resonance. We employ a $^{39}$K condensate trapped in an
optical lattice, where Bloch oscillations are forced by gravity. With a control
of the scattering length better that 0.1 $a_0$ we achieve coherence times of
several hundreds of ms. The micrometric sizes of the atomic sample make our
sensor an ideal candidate for measuring forces with high spatial resolution.
Our technique can be in principle extended to other measurement schemes opening
new possibilities in the field of trapped atom interferometry.
