We determine structure constants for the universal nonassociative enveloping algebra U(M) of the four-dimensional non-Lie Malcev algebra M by constructing a representation of U(M) by differential operators on the polynomial algebra P (M). These structure constants involve Stirling numbers of the second kind. This work is based on the recent theorem of Pérez-Izquierdo and Shestakov which generalizes the Poincaré-Birkhoff-Witt theorem from Lie algebras to Malcev algebras. We use our results for U(M) to determine structure constants for the universal alternative enveloping algebra A(M) = U(M)/I(M) where I(M) is the alternator ideal of U(M). The structure constants for A(M) were obtained earlier by Shestakov using different methods.