  We give a complete classification of intertwining operators (symmetry
breaking operators) between spherical principal series representations of
G=O(n+1,1) and G'=O(n,1). We construct three meromorphic families of the
symmetry breaking operators, and find their distribution kernels and their
residues at all poles explicitly.Symmetry breaking operators at exceptional
discrete parameters are thoroughly studied.
  We obtain closed formulae for the functional equations which the composition
of the the symmetry breaking operators with the Knapp-Stein intertwining
operators of $G$ and G' satisfy, and use them to determine the symmetry
breaking operators between irreducible composition factors of the spherical
principal series representations of G and G'.
  Some applications are included.
