EX Hya is one of the best studied, but still enigmatic intermediate polars. We present phase-resolved blue VLT/UVES high-resolution ({lambda}/{DELTA}{lambda}~=16.000) spectra of EX Hya taken in January 2004. Our analysis involves a unique decomposition of the Balmer line profiles into the spin-modulated line wings that represent streaming motions in the magnetosphere and the orbital-phase modulated line core that represents the accretion disk. Spectral analysis and tomography show that the division line between the two is solidly located at |v_rad_|~=1200km/s, defining the inner edge of the accretion disk at r_in_=~7x10^9^cm or ~10R1 (WD radii). This large central hole allows an unimpeded view of the tall accretion curtain at the lower pole with a shock height up to h_sh_~1R1 that is required by X-ray and optical observations. Our results contradict models that advocate a small magnetosphere and a small inner disk hole. Equating rin with the magnetospheric radius in the orbital plane allows us to derive a magnetic moment of the WD of {mu}1~=1.3x10^32^G.cm^3^ and a surface field strength B1~0.35MG. Given a polar field strength Bp~<1.0MG, optical circular polarization is not expected. With an accretion rate dM/dt=3.9x10^-11^M_{sun}/yr, the accretion torque is Gacc~=2.2x10^33^g.cm^2^/s^2^. The magnetostatic torque is of similar magnitude, suggesting that EX Hya is not far from being synchronized. We measured the orbital radial-velocity amplitude of the WD, K1=58.7+/-3.9km/s, and found a spin-dependent velocity modulation as well. The former is in perfect agreement with the mean velocity amplitude obtained by other researchers, confirming the published component masses M1~=0.79M{sun} and M2~=0.11M{sun}_ .