Experimental results are presented for the case of a sphere falling under gravity along the axis of a cylindrical tube containing an elastic liquid. The radius of the container is twice that of the sphere. The work is a logical extension of the earlier experiments of Jones et al. [W.M. Jones, A.H. Price and K. Waiters, J. Non-Newtonian Fluid Mech., 53 (1994) 175-196] and Degand and Waiters [E. Degand and K. Waiters, J. Non-Newtonian Fluid Mech., 53 (1995) 103-115], the new feature being the substantial shear thinning exhibited by the two polymer solutions investigated. It is found that, in general terms, shear thinning has a much stronger influence on the settling viscosity than the fluid elasticity, in line with the earlier conclusions of Mena et al. [B. Mena, O. Menero and L.G. Leal, J. Non-Newtonian Fluid Mech., 26 (1987) 247-275]. When an attempt is made to accommodate the shear thinning using a numerical simulation for an inelastic generalized Newtonian model, the generic behaviour of the normalized drag against Weissenberg number is not inconsistent with that postulated by Waiters and Tanner [K. Waiters and R.I. Tanner, in R.P. Chhabra and D. Dekee (Eds.), Transport Process in Bubbles, Drops and Particles, Hemisphere, 1991, pp. 73-86], i.e. a drag decrease at low Weissenberg numbers followed by a drag increase at high Weissenberg numbers.