  We show that four-dimensional Einstein-Maxwell-Dilaton-Gauss-Bonnet gravity
admits asymptotically flat black hole solutions with a degenerate event horizon
of the Reissner-Nordstr\"om type $AdS_2\times S^2$. Such black holes exist for
the dilaton coupling constant within the interval $0\leq a^2<a^2_{\rm cr}$.
Black holes must be endowed with an electric charge and (possibly) with
magnetic charge (dyons) but they can not be purely magnetic. Purely electric
solutions are constructed numerically and the critical dilaton coupling is
determined $a_{\rm cr}\simeq 0.488219703$. For each value of the dilaton
coupling $a$ within this interval and for a fixed value of the Gauss--Bonnet
coupling $\alpha$ we have a family of black holes parameterized by their
electric charge. Relation between the mass, the electric charge and the dilaton
charge at both ends of the allowed interval of $a$ is reminiscent of the BPS
condition for dilaton black holes in the Einstein-Maxwell-Dilaton theory. The
entropy of the DGB extremal black holes is twice the Bekenstein-Hawking
entropy.
