Microtubules of the mitotic spindle form the structural basis for chromosome segregation. In metaphase, microtubules show high dynamic instability, which is thought to aid the 'search and capture' of chromosomes for bipolar alignment on the spindle. Microtubules suddenly become more stable at the onset of anaphase, but how this change in microtubule behaviour is regulated and how important it is for the ensuing chromosome segregation are unknown. Here we show that in the budding yeast Saccharomyces cerevisiae, activation of the phosphatase Cdc14 at anaphase onset is both necessary and sufficient for silencing microtubule dynamics. Cdc14 is activated by separase, the protease that triggers sister chromatid separation, linking the onset of anaphase to microtubule stabilization. If sister chromatids separate in the absence of Cdc14 activity, microtubules maintain high dynamic instability; this correlates with defects in both the movement of chromosomes to the spindle poles (anaphase A) and the elongation of the anaphase spindle (anaphase B). Cdc14 promotes localization of microtubule-stabilizing proteins to the anaphase spindle, and dephosphorylation of the kinetochore component Ask1 contributes to both the silencing of microtubule turnover and successful anaphase A.