We present the first direct experimental bound on the total decay width of the top quark, Γt, using 955  pb−1 of the Tevatron's pp collisions recorded by the Collider Detector at Fermilab. We identify 253 top-antitop pair candidate events. The distribution of reconstructed top quark mass from these events is fitted to templates representing different values of the top quark width. Using a confidence interval based on likelihood-ratio ordering, we extract an upper limit at 95% C.L. of Γt<13.1  GeV for an assumed top quark mass of 175  GeV/c2.