  Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$
generated by higher integrals of the quantum Gaudin magnet chain attached to a
semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of
pairwise distinct complex numbers $z_1,\ldots,z_N$. We prove that this
subalgebra has a cyclic vector in the space of singular vectors of the tensor
product of any finite-dimensional irreducible $\mathfrak{g}$-modules, for all
values of the parameters $z_1,\ldots,z_N$. We deduce from this result the Bethe
Ansatz conjecture in the Feigin-Frenkel form which states that the joint
eigenvalues of the higher Gaudin Hamiltonians on the tensor product of
irreducible finite-dimensional $\mathfrak{g}$-modules are in 1-1 correspondence
with monodromy-free ${}^LG$-opers on the projective line with regular
singularities at the points $z_1,\ldots,z_N,\infty$ and the prescribed residues
at the singular points.
