1 To any d-web of codimension one on a holomorphic n-dimensional manifold M ( d > n ), we associate an analytic subset S of M. We call regular the webs for which S has at most dimension n − 1 . This condition is generically satisfied. 1 We changed recently the terminology, and call now “ordinary” instead of “regular” the webs which are studied in this note. Denoting by c ( n , h ) the dimension of the vector space of homogeneous polynomials of degree h in n variables, we prove that the rank of a regular web has an upper-bound π ′ ( n , d ) equal to 0 for d < c ( n , 2 ) , and to ∑ h = 1 k 0 ( d − c ( n , h ) ) for d ⩾ c ( n , 2 ) , k 0 denoting the integer such that c ( n , k 0 ) ⩽ d < c ( n , k 0 + 1 ) . This bound π ′ ( n , d ) is optimal for regular webs. For n ⩾ 3 , it is strictly smaller than the bound π ( n , d ) of Chern–Castelnuovo. Moreover, if d is precisely equal to c ( n , k 0 ) , we define a holomorphic connection on some vector bundle E of rank π ′ ( n , d ) above M ∖ S , such that the vector space of germs of Abelian relation of the web at a point of M ∖ S is isomorphic to the vector space of germs at that point of holomorphic sections of E with vanishing covariant derivative: the curvature of this connection, which generalizes the curvature of Blaschke–Dubourdieu–Panzani–Hénaut, is then the obstruction for the rank of the web to reach the value π ′ ( n , d ) . [When n = 2 , S is always empty so that any web is regular, π ′ ( 2 , d ) = π ( 2 , d ) , any d may be written c ( 2 , k 0 ) : we recover the results given in Pantazi (1938) and Hénaut (2004).] To cite this article: V. Cavalier, D. Lehmann, C. R. Acad. Sci. Paris, Ser. I 346 (2008).