A \(W^*\)-subalgebra \(\mathcal A_0\) of a \(W^*\)-algebra \(\mathcal A\) is said to be normal in \(\mathcal A\) if \((\mathcal A'_0\cap \mathcal A)' \cap \mathcal A = \mathcal A_0\) (i.e., if \(\mathcal A_0\) has the double commutant property relative to \(\mathcal A)\). [A \(W^*\)-algebra \(\mathcal A\) is normal if every \(W^*\)-subalgebra \(\mathcal A_0\) of \(\mathcal A\) containing the center of \(\mathcal A\) is normal in \(\mathcal A\). All type I \(W^*\)-algebras are normal. No type II factor is normal.] A few examples of non-normal abelian subalgebras in factors of type II have appeared in the literature [\textit{J. Dixmier}, Ann. Math. (2) 59, 279--286 (1954; Zbl 0055.10702); \textit{R. Kadison}, Duke Math. J. 29, 459--464 (1962; Zbl 0177.17802); \textit{T. Saitô}, Tôhoku Math. J. (2) 17, 206--209 (1965; Zbl 0135.17401)]. In this paper there is constructed an infinite sequence of abelian subalgebras which are non-normal in a hyperfinite type \(\mathrm{II}_1\) factor \(\mathcal A\) and which are pairwise non-conjugate under \(*\)-automorphisms of \(\mathcal A\).