  Deligne's regularity criterion for an integrable connection $\nabla$ on a
smooth complex algebraic variety $X$ says that $\nabla$ is regular along the
irreducible divisors at infinity in some fixed normal compactification of $X$
if and only if the restriction of $\nabla$ to every smooth curve on $X$ is
regular ({\it i. e.} has only regular singularities at infinity). The ``only
if" part is the difficult implication. Deligne's proof is transcendental, and
uses Hironaka's resolution of singularities. We give here an elementary and
purely algebraic proof of this implication: it is, as far as we know, the first
algebraic proof of Deligne's regularity criterion.
