An alumina sorbent capable of adsorbing NOx and SOx from waste gases and being regenerated by heating above 600.degree. C. is made by incorporating an alumina stabilizing agent into the sorbent. A preferred method is to add the stabilizer when the alumina is precipitated. The precipitated powder is formed subsequently into a slurry, milled and dripped to form the stabilized spheroidal alumina particles. These particles are impregnated with an alkali metal or alkaline earth metal to form the stabilized sorbent. Alumina stabilizers include one or more of silica, lanthana, other rare earths, titania, zirconia and alkaline earths.