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We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlinmodules. The uniqueness result leads to a complete classification of theirreducible Gelfand-Tsetlin modules with 1-singularity. An explicitconstruction of such modules was given in \cite{FGR2}. In particular, we showthat the modules constructed in \cite{FGR2} exhaust all irreducibleGelfand-Tsetlin modules with 1-singularity. To prove the result we introduce anew category of modules (called Drinfeld category) related to the Drinfeldgenerators of the Yangian Y(gl_n) and define a functor from the category ofnon-critical Gelfand-Tsetlin modules to the Drinfeld category.

Drinfeld category and the classification of singular Gelfand-Tsetlin gl_n-modules