The manifestly SU(4)xU(1) super-Poincaré 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, ∂ z 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 ∂ z x + , these amplitudes are shown to agree with amplitudes obtained using the manifestly unitary light-cone gauge formalism.