  Aims. Using ground based multi-object optical spectroscopy obtained in the
framework of the VLT-FLAMES Tarantula Survey (VFTS), we aim to establish the
(projected) rotational velocity distribution for a sample of 216 presumably
single O-type stars in 30 Doradus (30 Dor).
  Methods. We measured projected rotational velocities, \vrot, by means of a
Fourier transform method and a profile fitting method applied on a set of
isolated spectral lines. We also used an iterative deconvolution procedure to
infer the probability density, $\rm{P(\veq)}$, of the equatorial rotational
velocity, \veq.
  Results. The distribution of \vrot\ shows a two-component structure: a peak
around 80 \kms\ and a high-velocity tail extending up to $\sim$600 \kms. This
structure is also present in the inferred distribution $\rm{P(\veq)}$ with
around 80% of the sample having 0 $<$ \veq\, $\leq\, 300$ \kms\ and the other
20% distributed in the high-velocity region.
  Conclusions. Most of the stars in our sample rotate with a rate less than
20%\ of their break-up velocity. For the bulk of the sample, mass-loss in a
stellar wind and/or envelope expansion is not efficient enough to significantly
spin down these stars within the first few Myr of evolution. The presence of a
sizeable population of fast rotators is compatible with recent population
synthesis computations that investigate the influence of binary evolution on
the rotation rate of massive stars. Despite the fact that we have excluded
stars that show significant radial velocity variations, our sample may have
remained contaminated by post-interaction binary products. The fact that the
high-velocity tail may be preferentially (and perhaps even exclusively),
populated by post-binary interaction products, has important implications for
the evolutionary origin of systems that produce gamma-ray bursts.
