  When M-theory is compactified on G_2-holonomy manifolds with conical
singularities, charged chiral fermions are present and the low-energy
four-dimensional theory is potentially anomalous. We reconsider the issue of
anomaly cancellation, first studied by Witten. We propose a mechanism that
provides local cancellation of all gauge and mixed gauge-gravitational
anomalies, i.e. separately for each conical singularity. It is similar in
spirit to the one used to cancel the normal bundle anomaly in the presence of
five-branes. It involves smoothly cutting off all fields close to the conical
singularities, resulting in an anomalous variation of the 3-form C and of the
non-abelian gauge fields present if there are also ADE singularities.
