For \(\Re s>1\) and a fixed parameter \(\alpha\in(0,1]\), the Hurwitz zeta-function is given by \(\zeta(s,\alpha)=\sum_{m=0}^\infty (m+\alpha)^{-s}\) and by analytic continuation elsewhere except for a simple pole at \(s=1\). In the present paper the author proves a limit theorem in the space of analytic functions (in the sense of weakly convergent probability measures) for \(\zeta(s,\alpha)\) with \(\alpha\) algebraic irrational. This continues the author's studies on the value-distribution of these and other generalizations of the Riemann zeta-function; for a nice survey see [A survey on limit theorems for general Dirichlet series, Fiz. Mat. Fak. Moksl. Semin. Darb. 7, 45--56; (2004; Zbl 1104.11044)]. There is some hope that this limit theorem might be of some help to prove the universality for \(\zeta(s,\alpha)\) with \(\alpha\) algebraic irrational (which is an open problem).