  Very metal-poor stars that have $[\text{Fe}/\text{H}]<-2$ and that are
enhanced in C relative to Fe ($[\text{C}/\text{Fe}]>+0.7$) but have no
enhancement of heavy elements ($[\text{Ba}/\text{Fe}]<0$) are known as
carbon-enhanced metal-poor (CEMP-no) stars. These stars are thought to be
produced from a gas that was polluted by the supernova (SN) ejecta of the very
first generation (Pop III) massive stars. The very high enrichment of C
($A(\text{C})\gtrsim 6$) observed in many of the CEMP-no stars is difficult to
explain by current models of SN explosions from massive Pop III stars when a
reasonable dilution of the SN ejecta, that is consistent with detailed
simulation of metal mixing in minihaloes, is adopted. We explore rapidly
rotating Pop III stars that undergo efficient mixing and reach a
quasi-chemically homogeneous (QCH) state. We find that QCH stars can eject
large amounts of C in the wind and that the resulting dilution of the wind
ejecta in the interstellar medium can lead to a C enrichment of
$A(\text{C})\lesssim7.75$. The core of QCH stars can produce up to an order of
magnitude of more C than non-rotating progenitors of similar mass and the
resulting SN can lead to a C enrichment of $A(\text{C})\lesssim7$. Our rapidly
rotating massive Pop III stars cover almost the entire range of $A(\text{C})$
observed in CEMP-no stars and are a promising site for explaining the high C
enhancement in the early Galaxy. Our work indicates that a substantial fraction
of Pop III stars were likely rapid rotators.
