  Let $f: X \rightarrow C$ be a genus 1 fibration from a smooth projective
surface, i.e. its generic fiber is a regular genus 1 curve. Let $j: J
\rightarrow C$ be the Jacobian fibration of $f$. In this paper, we prove that
the Chow motives of $X$ and $J$ are isomorphic. As an application, combined
with our concomitant work on motives of quasi-elliptic fibrations, we prove
Kimura finite-dimensionality for smooth projective surfaces not of general type
with geometric genus 0. This generalizes Bloch-Kas-Lieberman's result to
arbitrary characteristic.
