  We prove an index theorem for boundary value problems in Boutet de Monvel's
calculus on a compact manifold X with boundary. The basic tool is the tangent
semigroupoid $\cT^-X$ generalizing the tangent groupoid defined by Connes in
the boundaryless case, and an associated continuous field C*_r(\cT^-X) of
C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with
the symbol algebra for Boutet de Monvel's calculus; for h\not=0 the fibers are
isomorphic to the algebra K of compact operators. We therefore obtain a natural
map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show
that this is the analytic index map. On the other hand, using ideas from
noncommutative geometry, we construct the topological index map and prove that
it coincides with the analytic index map.
