A high-frequency torsional rod apparatus (HFTRA) has been developed for measurements of moderately high-frequency viscoelastic properties of dilute, low viscosity polymer solutions. It employs a long cylindrically shaped resonating element driven by an X-cut (torsional mode) quartz piezoelectric crystal. Up to 11 different normal modes are employed, resulting in 11 discrete frequencies, ranging from approximately 20 to 500 kHz. Measurements are made in free decay; a key feature of the apparatus is the method for high precision measurements of damping coefficients and eigenfrequencies. The instrument is suitable for liquids with 0.003P < eta* < 5P and eta’ greater-than-or-equal-to 1.33 eta". Typically, eta’ has a relative uncertainty of +/- 2% or less, with the relative uncertainty for eta" being about 4% for liquids with eta" greater than 0.01P. Values of eta’ obtained in the HFTRA for low viscosity liquids with small eta" show excellent agreement with the known steady flow viscosities. Measurements of the viscoelastic properties of polymer solutions employing the HFTRA and two other instruments capable of higher precision measurements show excellent agreement, demonstrating that the HFTRA is sufficiently precise for dilute-solution viscoelasticity studies; the working frequency range is such that most of the polymer relaxation time spectrum can be probed for many polymer/solvent combinations.