  The manifestly SU(4)xU(1) super-Poincare invariant free-field N=2 twistor-
string action for the ten-dimensional Green-Schwarz superstring is quantized
using standard BRST methods. Unlike the light-cone and semi-light-cone
gauge-fixed Green-Schwarz actions, the twistor-string action does not require
interaction-point operators at the zeroes of the light-cone momentum, $\dz
x^+$, which complicated all previous calculations. After defining the vertex
operator for the massless physical supermultiplet, as well as two
picture-changing operators and an instanton-number-changing operator,
scattering amplitudes for an arbitrary number of loops and external massless
states are explicitly calculated by evaluating free-field correlation functions
of these operators on N=2 super-Riemann surfaces of the appropriate topology,
and integrating over the global moduli. Although there is no sum over spin
structures, only discrete values of the global U(1) moduli contribute to the
amplitudes. Because the spacetime supersymmetry generators do not contain ghost
fields, the amplitudes are manifestly spacetime supersymmetric, there is no
multiloop ambiguity, and the non-renormalization theorem is easily proven. By
choosing the picture-changing operators to be located at the zeroes of $\dz
x^+$, these amplitudes are shown to agree with amplitudes obtained using the
manifestly unitary light-cone gauge formalism.
