The absorption of electron cyclotron waves in hot and dense plasmas is investigated by using the relativistic dispersion relation with finite Larmor radius (FLR) corrections. It is shown that the strong reduction caused by FLR effects on the damping of extraordinary waves propagating perpendicularly to the magnetic field becomes insignificant at large values of the parallel component of the refractive index. An application of these results to the auxiliary heating of dense plasmas of a compact ignition tokamak [PlasmaPhysicsandControlledNuclearFusion1986 (IAEA, Vienna, 1987), Vol. 3] is given. It is shown that the extraordinary mode with oblique propagation and frequency of 190 GHz can be strongly absorbed by dense plasmas with a toroidal magnetic field of 105 kG and a central electron density of 1×1015cm−3.