We used optical spectroscopy of the neutron star X-ray transient XTE J2123-058 to measure the rotational broadening of the binary companion's stellar absorption lines and determined that the companion's projected rotational velocity is v sin i = 121(-29)(+21) km s(-1). This value is considerably larger than the measurements of v sin i obtained previously for three other neutron star systems, where the values were between 34 and 65 km s(-1). The larger value is likely due to the combination of high binary inclination and short ( 6 hr) orbital period for XTE J2123-058. Along with previously measured parameters, the rotational broadening measurement allowed us to determine the binary parameters, including the ratio of the neutron star mass to the companion mass, q = M-1/M-2 = 2.7(-0.9)(+1.3) the neutron star mass M-1 = 1.46(-0.39)(+0.30) M-., and the companion mass M-2 = 0.53(-0.39)(+0.28) M-., assuming a Roche lobe-filling companion synchronously rotating with the binary orbit. These values are consistent with a previous measurement in which the H emission line was used to determine the semiamplitude of the neutron star's radial velocity curve (K-1). It is a significant result that the two methods give consistent values. We also report the first measurement of the XTE J2123-058 companion's radius. Assuming synchronous rotation, we obtain R-2 = 0.62(-0.15)(+0.11) R-., which is in line with that expected for a late K-type star on or close to the main sequence. Finally, we report the first precise spectroscopic determination of the binary orbital period (P-orb = 21, 442.3 +/- 1.8 s).