By an equicontinuous topological dynamical system \((X,T)\) is understood a pair consisting of a compact Hausdorff space \(X\) and a homeomorphism \(T: X\to X\) such that for every continuous function \(f: X\to\mathbb{R}\) the family \(\{f\circ T^ n: n\in\mathbb{Z}\}\) is equicontinuous. For the system \((X,T)\) we define the function \(P: X\to\mathbb{N}\cup\{\infty\}\) setting \(P(x)=n\in\mathbb{N}\) if the point \(x\) is periodic with minimal period \(n\) and \(P(x)=\infty\) if \(x\) is aperiodic. Denoting by \(S(X,T)\) the range of the function \(P\) the author gives a complete characterization of this set in the case when \(\infty\in S(X,T)\), proving the following statement: A set \(S\subset \mathbb{N}\cup\{\infty\}\), such that \(\infty\in S\) is the set of periods of an equicontinuous system if and only if \(S={\overset{.}\bigcup}^ \infty_{i=1} S_ i\cup\{\infty\}\), where \(S_ i\) is a 1-based set and the following condition holds: (1) for every \(q\geq 0\) there exist \(p_ 1,p_ 2,\dots,p_ k\geq q\) and \(n\) such that for every \(i\geq n\), \(p_ j| S_ i\) for some \(j\), \(1\leq j\leq k\). Here \({\overset{.}\bigcup}\) denotes the disjoint union of sets, \(p| S_ i\) denotes that \(p\) divides every element of the set \(S_ i\) and \(S_ i\) being a 1-based set means that there exists a number \(a_ i\) such that every element of \(S_ i\) is a multiple of \(a_ i\).