Single-spin asymmetries for semi-inclusive electroproduction of charged pions in deep-inelastic scattering of positrons are measured for the first time with transverse target polarization. The asymmetry depends on the azimuthal angles of both the pion (phi) and the target spin axis (phi(S)) about the virtual-photon direction and relative to the lepton scattering plane. The extracted Fourier component sin((phi+phi(S))(pi)(UT) is a signal of the previously unmeasured quark transversity distribution, in conjunction with the Collins fragmentation function, also unknown. The component sin((phi-phi(S)(pi)(UT) arises from a correlation between the transverse polarization of the target nucleon and the intrinsic transverse momentum of quarks, as represented by the previously unmeasured Sivers distribution function. Evidence for both signals is observed, but the Sivers asymmetry may be affected by exclusive vector meson production.