In this paper we prove the Knop conjecture asserting that two smooth affine
spherical varieties with the same weight monoids are equivariantly isomorphic.
We also state and prove a uniqueness property for not necessarily smooth affine
spherical varietiesComment: 31 pages, v2,v3 typos fixed, minor mistakes are corrected, references
added v4 20 pages, crucial changes made. The proof now uses results of
AG/0703543. A new result concerning a uniqueness property of not necessarily
smooth affine spherical varieties is added, v6 final version, to appear in
Annales de l Institut Fourie