The {sup 24}Mg({sup 3}He,t){sup 24}Al reaction was studied at E({sup 3}He)=420 MeV. An energy resolution of 35 keV was achieved. Gamow-Teller strengths to discrete levels in {sup 24}Al were extracted by using a recently developed empirical relationship for the proportionality between Gamow-Teller strengths and differential cross sections at zero momentum transfer. Except for small discrepancies for a few weak excitations, good agreement with previous {sup 24}Mg(p,n) data and nuclear-structure calculations using the USDA/B interactions in the sd shell-model space was found. The excitation energy of several levels in {sup 24}Al of significance for determination of the {sup 23}Mg(p,{gamma}){sup 24}Al thermonuclear reaction rate were measured. Results are consistent with two of the three previous ({sup 3}He,t) measurements, performed at much lower beam energies. However, a new state at E{sub x}({sup 24}Al)=2.605(10) MeV was found and is the third state above the proton separation energy.