The authors consider the method of Tikhonov regularization in Hilbert scales for finding solutions \(x^*\) of the ill-posed linear and nonlinear operator equation \(A(x)= z\), where \(A: X\to Y\) is an operator between two Hilbert spaces. The method consists of minimizing the functional \[ J_ \alpha(x):= \| A(x)- z^ \delta\|^ 2+ \alpha\| x- \bar x\|^ 2_ s,\quad \alpha>0, \] where \(\| z^ \delta- z\|\leq \delta\) and \(\|\cdot\|_ s\) is the norm in a Hilbert scale \(X_ s\). Assuming \(\| A(x)- A(x^*)\|\sim \| x- x^*\|_{-a}\) and \(x^*- x\in X_{2\gamma}\) for some \(a\geq 0\) and \(\gamma\geq 0\), they prove that the choice of the regularization parameter \(\alpha\) by Morozov’s discrepancy principle leads to optimal convergence rates if \(2\gamma\in [s,2s+ a]\). They also discuss convergence rate results for the case when \(2\gamma\) in this interval and apply the results to linear integral equations.