Existing constitutive theories for fibre suspensions in Newtonian suspending media adequately predict the steady shear and extensional viscosity for such dilute and semi-dilute suspensions, but most do not predict the finite normal stresses observed for non-dilute fibres suspended in a Newtonian liquid. Normal stresses were measured for suspensions of rigid glass fibres in Newtonian fluids. The fibres were 3.2 mm (1/8'') and 6.3 mm (1/4'') long, having aspect ratios of 276 and 552 respectively. The first normal stress difference was found to vary approximately linearly with shear rate and its magnitude increased with increasing fibre concentration and aspect ratio. The fact that these observed normal stresses are not merely due to wall effects was confirmed by varying the gap in a parallel-plate geometry for the normal stress measurements. It was found that the normal stresses observed in steady state shear were adequately predicted by the equation derived by Carter [L.F. Carter, A Study of the Rheology of the Suspensions of Rod-Shaped Particles in a Navier-Stokes liquid, Ph.D. Dissertation, University of Michigan, Ann Arbor, MI, 1967].