  We study the stability under scalar perturbations, and we compute the
quasinormal modes of the Einstein-Born-Infeld dilaton spacetime in 1+3
dimensions. Solving the full radial equation in terms of hypergeometric
functions, we provide an exact analytical expression for the spectrum. We find
that the frequencies are purely imaginary, and we confirm our results by
computing them numerically. Although the scalar field that perturbs the black
hole is electrically neutral, an instability similar to that seen in charged
scalar perturbations of the Reissner-Nordstr\"om black hole is observed.
