Summary: We consider a one-dimensional transient cookie random walk. It is known from a previous paper [A.-L. Basdevant and A. Singh, Probab. Theory Relat. Fields 141, No. 3–4, 625–645 (2008; Zbl 1141.60383)] that a cookie random walk \((X_n)\) has positive or zero speed according to some positive parameter \(\alpha >1\) or \(\leq 1\). In this article, we give the exact rate of growth of \(X_n\) in the zero speed regime, namely: for \(0<\alpha <1, X_{n}/n^{(\alpha +1)/2}\) converges in law to a Mittag-Leffler distribution whereas for \(\alpha =1, X_n(\log n)/n\) converges in probability to some positive constant.