  We derive, using the pure-spinor formalism, the complete -- including the
fermions -- four-point effective action of both type II superstrings to all
orders in $\alpha'$, at tree level in string loops. We find that, in the
quartic-field approximation, the supergravity Lagrangian can be thought of as
the tensor product, in a suitable sense, of two copies of the superYang-Mills
Lagrangian in ten dimensions. The NS-NS three-form enters the supergravity
Lagrangian through a modified connection with torsion. As a byproduct, we
derive the complete, i.e. to all orders in the $\theta$-expansion,
closed-string vertex operator in a flat target-space background.
