The author constructs an associative algebra A by a two-fold limit procedure taking first the projective limit \(A_ m\) of suitably chosen subalgebras \(A_ m(n)\) of the enveloping algebras U(\({\mathfrak gl}(n))\), and then the direct limit with respect to \(m\to \infty\). The algebra A has a nontrivial center \(A_ 0\) and may be used instead of the enveloping algebra in case of the infinite-dimensional Lie groups SO(\(\infty)\) and Sp(\(\infty)\). \(A_ 0\) is isomorphic to the algebra of Laplace operators. It is proved, that \(A_ m\) is isomorphic to the tensor product of \(A_ 0\) with the Yangian Y(\({\mathfrak gl}(m))\), which implies an isomorphism of A with the tensor product of \(A_ 0\) with the Yangian Y(\({\mathfrak gl}(\infty))\). Moreover every irreducible highest weight \({\mathfrak gl}(\infty)\)-module is an A-module.