Aims: Projected rotational velocities (vesini) 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 km s-1 and show a bi-modal structure. This is also present in the inferred rotational velocity distribution with 25% of the sample having 0 ≤ ve ≤ 100 km s-1 and the high velocity component having ve ~ 250 km s-1. 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. Tables 3 and 4 are only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/550/A109