We consider the effect of disorder on coherent tunneling through two barriers in series, in the regime of overlapping transmission resonances. We present analytical calculations (using random-matrix theory) and numerical simulations (on a lattice) to show that strong mode-mixing in the inter-barrier region induces mesoscopic fluctuations in the conductance $G$ of universal magnitude $e^2/h$ for a symmetric junction. For an asymmetric junction, the root-mean-square fluctuations depend on the ratio $ν$ of the two tunnel resistances according to ${rms} G = (4e^2/h)β^{-1/2} ν(1+ν)^{-2}$, where $β= 1 (2)$ in the presence (absence) of time-reversal symmetry.

12 pages, REVTeX-3.0, 2 figures, submitted to Physical Review B