  We construct the smeared diffeomorphism constraint operator at finite
triangulation from the basic holonomy- flux operators of Loop Quantum Gravity,
evaluate its continuum limit on the Lewandowski- Marolf habitat and show that
the action of the continuum operator provides an anomaly free representation of
the Lie algebra of diffeomorphisms of the 3- manifold. Key features of our
analysis include: (i) finite triangulation approximants to the curvature,
$F_{ab}^i$ of the Ashtekar- Barbero connection which involve not only small
loop holonomies but also small surface fluxes as well as an explicit dependence
on the edge labels of the spin network being acted on (ii) the dependence of
the small loop underlying the holonomy on both the direction and magnitude of
the shift vector field (iii) continuum constraint operators which do {\em not}
have finite action on the kinematic Hilbert space, thus implementing a key
lesson from recent studies of parameterised field theory by the authors.
  Features (i) and (ii) provide the first hints in LQG of a conceptual
similarity with the so called "mu- bar" scheme of Loop Quantum Cosmology. We
expect our work to be of use in the construction of an anomaly free quantum
dynamics for LQG.
