Linear systems involving contiguous submatrices of the discrete Fourier transform (DFT)matrix arise in many applications, such as Fourier extension, superresolution, and coherent diffraction imaging. We show that the condition number of any such p\times q submatrix of the N\times NDFT matrix is at least exp\bigl( \pi 2\bigl[ min(p,q) - pqN\bigr] \bigr) , up to algebraic prefactors.That is, fixing the shape parameters (\alpha ,\beta ) := (p/N,q/N)\in (0,1)2, the growth ise\rho NasN\rightarrow \infty , the exponential rate being\rho =\pi 2[min(\alpha ,\beta ) - \alpha \beta ]. Our proof uses theKaiser--Bessel transform pair (of which we give a self-contained proof), plus estimates on sums over distorted sinc functions, to construct a localized trial vector whose DFT is also localized. We warm up with an elementary proof of the above but with half the rate, via a periodized Gaussian trial vector. Using low-rank approximation of the kerneleixt, we also prove another lower bound (4/e\pi \alpha )q, up to algebraic prefactors, which is stronger than the above for small\alpha and\beta . When combined, the bounds are within a factor of two ofthe empirical asymptotic rate, uniformly over (0,1)2, and become sharp in certain regions.However, the results are not asymptotic: they apply to essentially allN,p, andq, and with all constants explicit.