A group \(G\) is said to be hypercentral if the ascending central series reaches \(G\) after some, possibly infinite, ordinal. Let \(U(KG)\) be the group of units of the group ring \(KG\) of the group \(G\) over a field \(K\). If \(K\) has characteristic \(p>0\) and \(G\) contains \(p\)-elements, then it is proved that \(U(KG)\) is hypercentral if and only if \(G\) is nilpotent and \(G'\) is a finite \(p\)-group.