We present a perfect simulation algorithm for stationary processes indexed by $\mathbb{Z}$, with summable memory decay. Depending on the decay, we construct the process on finite or semi-infinite intervals, explicitly from an i.i.d. uniform sequence. Even though the process has infinite memory, its value at time 0 depends only on a finite, but random, number of these uniform variables. The algorithm is based on a recent regenerative construction of these measures by Ferrari, Maass, Martínez and Ney. As applications, we discuss the perfect simulation of binary autoregressions and Markov chains on the unit interval.