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We present non-LTE analyses of high quality spectra for two B-type supergiants, κ Orionis & ε Orionis (spectral types B0.5Ia & B0Ia respectively). We investigate the effect of including a microturbulent velocity in the HI, HeII and HeI line broadening on the derived stellar atmospheric parameters (T_eff, logg and y). Its inclusion has a significant effect on the profiles of the lines of neutral helium and leads to approximately normal estimates for the helium fractional abundances for both supergiants. By contrast, adopting zero microturbulence implies significant helium overabundances, if we adopt the mean abundance of all the lines considered here. The HeI lines at 4437, 4387 and 5047Å are found to be rather insensitive to microturbulence and hence are more reliable indicators of helium abundance. There are some remaining unresolved discrepancies, such as a systematic difference between singlet and triplet transitions which we attribute to the neglect of line blocking, while the 5015Å 2¹S - 3¹P transition remains much stronger than predicted. This latter problem we tentatively attribute to the neglect of sphericity; the generalized dilution effect discussed by Voels et al. (1989). We suggest that a judicious choice of Hei lines, and the use of an appropriate microturbulent velocity, may contribute to resolving the `helium discrepancy' in O-type stars. The question of the origin of a perceived microturbulence in supergiants is also briefly discussed in the context of stellar winds. That two normal supergiants have close to solar helium abundances clearly implies that they have not undergone dredge-up in a previous red supergiant phase of evolution, supporting the contention that massive stars do not perform blue loops in the HR diagram at solar metallicity. Marginal evidence for a mild helium overabundance in κ Ori however could be interpreted as being due to mixing processes, perhaps during its main sequence lifetime.