Model liquids with nearly constant viscosity and adjustable elasticity are needed to resolve the role of elasticity in coating and other free-surface flows. Available Boger liquids are not well suited to free-surface flows, because they are solutions in organic solvents and their viscosities exceeding 1 Pa · s fall on the high side. Aqueous liquids are preferred in laboratory studies partly due to environmental hazards. Aqueous polymer solutions with constant shear viscosity and adjustable elasticity were prepared by adding small amounts of a high-molecular-weight polymer to a more concentrated aqueous solution of the same polymer but of a much lower molecular weight. Up to 0.2 wt. % of high-molecular weight poly(ethylene oxide) (PEO, Mw from 400,000 to 4 million g/mol) was added to almost inelastic solutions of low-MW polyethylene glycol (PEG, Mn = 8,000 g/mol). PEG concentrations in these solutions varied between 16.7 and 42.9 wt. %. Shear viscosities of these solutions ranged from about 0.02 to 0.3 Pa · s and were constant up to shear rates of 100 s−1. The stress ratio is one measure of the elasticity of the liquid. Stress ratios up to 0.2 were estimated from small-amplitude oscillatory measurements. Terminal behavior (elastic modulus rising with the square of the frequency) was not observed even at 10−2 rad/s. Viscosity and elasticity of the liquids can be manipulated over a wide range by varying the amounts and molecular weights of PEG and PEO within the unentangled and dilute regions of the concentration–molecular weight diagram, respectively. Fits of experimental data to candidate differential and integral constitutive equations are also discussed.