  Let g be an affine Lie algebra with associated Yangian Y_hg. We prove the
existence of two meromorphic R-matrices associated to any pair of
representations of Y_hg in the category O. They are related by a unitary
constraint and constructed as products of the form R(s)=R^+(s)R^0(s)R^-(s),
where R^+(s) = R^-_{21}(-s)^{-1}. The factor R^0(s) is a meromorphic, abelian
R-matrix, with a WKB-type singularity in h, and R^-(s) is a rational twist. Our
proof relies on two novel ingredients. The first is an irregular, abelian,
additive difference equation whose difference operator is given in terms of the
q-Cartan matrix of g. The regularization of this difference equation gives rise
to R^0(s) as the exponentials of the two canonical fundamental solutions. The
second key ingredient is a higher order analogue of the adjoint action of the
affine Cartan subalgebra of g on Y_hg. This action has no classical
counterpart, and produces a system of linear equations from which R^-(s) is
recovered as the unique solution. Moreover, we show that both operators give
rise to the same rational R-matrix on the tensor product of any two
highest-weight representations.
