  We introduce a scheme by which flat bands with higher Chern number $\vert
C\vert>1$ can be designed in ultracold gases through a coherent manipulation of
Bloch bands. Inspired by quantum-optics methods, our approach consists in
creating a "dark Bloch band" by coupling a set of source bands through resonant
processes. Considering a $\Lambda$ system of three bands, the Chern number of
the dark band is found to follow a simple sum rule in terms of the Chern
numbers of the source bands: $C_D\!=\!C_1+C_2-C_3$. Altogether, our dark-state
scheme realizes a nearly flat Bloch band with predictable and tunable Chern
number $C_D$. We illustrate our method based on a $\Lambda$ system, formed of
the bands of the Harper-Hofstadter model, which leads to a nearly flat Chern
band with $C_D\!=\!2$. We explore a realistic sequence to load atoms into the
dark Chern band, as well as a probing scheme based on Hall drift measurements.
Dark Chern bands offer a practical platform where exotic fractional quantum
Hall states could be realized in ultracold gases.
