  Aims: Projected rotational velocities (\vsini) have been estimated for 334
targets in the VLT-FLAMES Tarantula survey that do not manifest significant
radial velocity variations and are not supergiants. They have spectral types
from approximately O9.5 to B3. The estimates have been analysed to infer the
underlying rotational velocity distribution, which is critical for
understanding the evolution of massive stars.
  Methods: Projected rotational velocities were deduced from the Fourier
transforms of spectral lines, with upper limits also being obtained from
profile fitting. For the narrower lined stars, metal and non-diffuse helium
lines were adopted, and for the broader lined stars, both non-diffuse and
diffuse helium lines; the estimates obtained using the different sets of lines
are in good agreement. The uncertainty in the mean estimates is typically 4%
for most targets. The iterative deconvolution procedure of Lucy has been used
to deduce the probability density distribution of the rotational velocities.
  Results: Projected rotational velocities range up to approximately 450 \kms
and show a bi-modal structure. This is also present in the inferred rotational
velocity distribution with 25% of the sample having $0\leq$\ve$\leq$100\,\kms
and the high velocity component having \ve$\sim 250$\,\kms. There is no
evidence from the spatial and radial velocity distributions of the two
components that they represent either field and cluster populations or
different episodes of star formation. Be-type stars have also been identified.
  Conclusions: The bi-modal rotational velocity distribution in our sample
resembles that found for late-B and early-A type stars. While magnetic braking
appears to be a possible mechanism for producing the low-velocity component, we
can not rule out alternative explanations.
