  We report a detailed study of $^{11}$B and $^{27}$Al NMR spin-lattice
relaxation rates ($1/T_1$), as well as of $^{27}$Al Knight shift (K) of
Mg$_{1-x}$Al$_x$B$_2$, $0\leq x\leq 1$. The obtained ($1/T_1T$) and K vs. x
plots are in excellent agreement with ab initio calculations. This asserts
experimentally the prediction that the Fermi surface is highly anisotropic,
consisting mainly of hole-type 2-D cylindrical sheets from bonding $2p_{x,y}$
boron orbitals. It is also shown that the density of states at the Fermi level
decreases sharply on Al doping and the 2-D sheets collapse at $x\approx 0.55$,
where the superconductive phase disappears.
