It is proved that for every odd \(n\geq 665\) and every \(p>1\) coprime to n the group variety with defining identities \(x^{np}=1\), \([x^ n,y^ n]=1\) has unsolvable word problem. It is the first known example of a periodic variety of groups with this property. The proof is hard, depending on the well known results of Novikov and Adyan.