We use 106pb−1 of data collected with the Collider Detector at Fermilab to search for narrow-width, vector particles decaying to a top and an antitop quark. Model independent upper limits on the cross section for narrow, vector resonances decaying to tt¯ are presented. At the 95% confidence level, we exclude the existence of a leptophobic Z′ boson in a model of top-color-assisted technicolor with mass MZ′<480GeV/c2 for natural width Γ=0.012MZ′, and MZ′<780GeV/c2 for Γ=0.04MZ′.