We provide a census of the apparent stellar angular momentum within one effective radius of a volume-limited sample of 260 early-type galaxies (ETGs) in the nearby Universe, using the integral-field spectroscopy obtained in the course of the ATLAS3D project. We exploit the λR parameter (previously used via a constant threshold value of 0.1) to characterize the existence of two families of ETGs: slow rotators which exhibit complex stellar velocity fields and often include stellar kinematically distinct cores, and fast rotators which have regular velocity fields. Our complete sample of 260 ETGs leads to a new criterion to disentangle fast and slow rotators which now includes a dependency on the apparent ellipticity ε. It separates the two classes significantly better than the previous prescription and better than a criterion based on V/σ: slow rotators and fast rotators have λR lower and larger than, respectively, where kFS= 0.31 for measurements made within an effective radius Re. We show that the vast majority of ETGs are fast rotators: these have the regular stellar rotation, with aligned photometric and kinematic axes (Paper II of this series), include discs and often bars and represent 86 ± 2 per cent (224/260) of all ETGs in the volume-limited ATLAS3D sample. Fast rotators span the full range of apparent ellipticities from ε= 0 to 0.85, and we suggest that they cover intrinsic ellipticities from about 0.35 to 0.85, the most flattened having morphologies consistent with spiral galaxies. Only a small fraction of ETGs are slow rotators representing 14 ± 2 per cent (36/260) of the ATLAS3D sample of ETGs. Of all slow rotators, 11 per cent (4/36) exhibit two counter-rotating stellar disc-like components and are rather low-mass objects (Mdyn < 1010.5M⊙). All other slow rotators (32/36) appear relatively round on the sky (εe < 0.4), tend to be massive (Mdyn > 1010.5M⊙), and often (17/32) exhibit kinematically distinct cores. Slow rotators dominate the high-mass end of ETGs in the ATLAS3D sample, with only about one-fourth of galaxies with masses above 1011.5M⊙ being fast rotators. We show that the a4 parameter which quantifies the isophote's disciness or boxiness does not seem to be simply related to the observed kinematics, while our new criterion based on λR and ε is nearly independent of the viewing angles. We further demonstrate that the classification of ETGs into ellipticals and lenticulars is misleading. Slow and fast rotators tend to be classified as ellipticals and lenticulars, respectively, but the contamination is strong enough to affect results solely based on such a scheme: 20 per cent of all fast rotators are classified as ellipticals, and more importantly 66 per cent of all ellipticals in the ATLAS3D sample are fast rotators. Fast and slow rotators illustrate the variety of complex processes shaping galactic systems, such as secular evolution, disc instabilities, interaction and merging, gas accretion, stripping and harassment, forming a sequence from high to low (stellar) baryonic angular momentum. Massive slow rotators represent the extreme instances within the red sequence of galaxies which might have suffered from significant merging without being able to rebuild a fast-rotating component within one effective radius. We therefore argue for a shift in the paradigm for ETGs, where the vast majority of ETGs are galaxies consistent with nearly oblate systems (with or without bars) and where only a small fraction of them (less than 12 per cent) have central (mildly) triaxial structures.