  Three dedicated reactor experiments, Double Chooz, RENO and Daya Bay, have
recently performed a precision measurement of the third standard mixing angle
$\theta_{13}$ exploiting a multiple baseline comparison of $\nu_e \to \nu_e$
disappearance driven by the atmospheric mass-squared splitting. In this paper
we show how the same technique can be used to put stringent limits on the
oscillations of the electron neutrino into a fourth very light sterile species
(VLS$\nu$) characterized by a mass-squared difference lying in the range
[$10^{-3} - 10^{-1}$] eV$^2$. We present accurate constraints on the admixture
$|U_{e4}|^2$ obtained by a 4-flavor analysis of the publicly available reactor
data. In addition, we show that the estimate of $\theta_{13}$ obtained by the
combination of the three reactor experiments is rather robust and substantially
independent of the 4-flavor-induced perturbations provided that the new
mass-squared splitting is not too low ($\gtrsim 6 \times 10^{-3}$ eV$^2$). We
briefly comment on the possible impact of VLS$\nu$'s on the rest of the
neutrino oscillation phenomenology and emphasize their potential role in the
cosmological "dark radiation" anomaly.
