  This paper proves conjectures originating in the physics literature regarding
the intersection exponents of Brownian motion in a half-plane. For instance,
suppose that B and B' are two independent planar Brownian motions started from
distinct points in a half-plane H. Then, as $t \to \infty$, $$ P [ B[0,t] \cap
B'[0,t] = \emptyset, B[0,t] \cup B'[0,t] \subset H ] = t^{-5/3 + o(1)}.$$ The
proofs use ideas and tools developped by the authors in previous papers. We
prove that one of the stochastic Loewner evolution processes (with parameter 6,
that we will call SLE_6 and which has been conjectured to correspond to the
scaling limit of critical percolation cluster boundaries) satisfies the
``conformal restriction property''. We establish a generalization of Cardy's
formula (for crossings of a rectangle by a percolation cluster) for SLE_6, from
which the exact values of intersection exponents for SLE_6 followd. Since this
process satisfies the conformal restriction property, the Brownian intersection
exponents can be determined from the SLE_6 intersection exponents.
  Results about intersection exponents in the whole plane appear in subsequent
papers.
