We present predictions for the prompt-neutrino flux arising from the decay of charmed mesons and baryons produced by the interactions of high-energy cosmic rays in the Earth's atmosphere, making use of a QCD approach on the basis of the general-mass variable-flavor-number scheme for the description of charm hadroproduction at next-to-leading order, complemented by a consistent set of fragmentation functions. This same scheme is used for the description of charm hadroproduction at both the Tevatron and the Large Hadron Collider. We compare the theoretical results to those already obtained by our and other groups with different theoretical approaches. We provide comparisons with the experimental results obtained by the IceCube Collaboration and we discuss implications for parton distribution functions.