B chromosomes are extra, non-essential chromosomes present in addition to the normal complement of A chromosomes. Many species of cichlid fish in Lake Malawi carry a haploid, female-restricted B chromosome. Here we show that this B chromosome exhibits drive, with an average transmission rate of 70%. The offspring of B-transmitting females exhibit a strongly female-biased sex ratio. Genotyping of these offspring reveals the B chromosome carries a female sex determiner that is epistatically dominant to an XY system on linkage group 7. We suggest that this sex determiner evolved to enhance the meiotic drive of the B chromosome. This is some of the first evidence that female meiotic drive can lead to the invasion of new sex chromosomes solely to benefit the driver, and not to compensate for skewed sex ratios.