In the presence of direct trajectories connecting source and drain contacts, the conductance of a quantum dot may exhibit resonances of the Fano type. Since Fano resonances result from the interference of two transmission pathways, their lineshape (as described by the Fano parameter q) is sensitive to dephasing in the quantum dot. We show that under certain circumstances the dephasing time can be extracted from a measurement of q for a single resonance. We also show that q fluctuates from level to level, and calculate its probability distribution for a chaotic quantum dot. Our results are relevant to recent experiments by Goeres et al.