  This is the first part of a two-part paper describing a new concept of
separation of variables applied to the Clebsch integrable case of the Kirchhoff
equations. There are two principal novelties: 1) Separating coordinates are
constructed (not guessed) by solving the Kowalewski separability conditions. 2)
The quadratures represent an apparently new generalization of the standard
Jacobi inversion problem of algebraic geometry.
  Part I explains the Kowalewski separability conditions and their
implementation to the Clebsch case. It is shown that the new separating
coordinates lead to quadratures involving Abelian differentials on two
different non-hyperelliptic curves (of genus higher than the dimension of the
invariant tori).
  In Part II these quadratures are interpreted as a new generalization of the
standard Abel--Jacobi map, and a procedure of its inversion in terms of
theta-functions is worked out. The theta-function solution is different from
that found long time ago by F. K\"otter, since the theta-functions used in this
paper have different period matrix.
