  We calculate the equation of state of isospin-symmetric nuclear matter in the
three-loop approximation of chiral perturbation theory. The contributions to
the energy per particle $\bar E(k_f)$ from one- and two-pion exchange diagrams
are ordered in powers of the Fermi momentum $k_f$ (modulo functions of $k_f
/m_\pi$). It is demonstrated that, already at order ${\cal O}(k_f^4)$, two-pion
exchange produces realistic nuclear binding. The underlying saturation
mechanism is surprisingly simple (in the chiral limit), namely the combination
of an attractive $k_f^3$-term and a repulsive $k_f^4$-term. The empirical
saturation point and the nuclear compressibility $K\simeq 250 $MeV are well
reproduced at order ${\cal O}(k_f^5)$ with a momentum cut-off of $\Lambda
\simeq 0.65$ GeV which parametrizes short-range dynamics. No further
short-distance terms are required in our calculation of nuclear matter. In the
same framework we calculate the density-dependent asymmetry energy and find
$A_0\simeq 34 $MeV at the saturation point, in good agreement with the
empirical value. The pure neutron matter equation of state is also in fair
qualitative agreement with sophisticated many-body calculations and a
resummation result of effective field theory, but only for low neutron
densities $\rho_n <0.25 $fm$^{-3}$.
