Cygnus X-2 is one of the brightest and longest known X-ray sources. We present high-resolution optical spectroscopy of Cyg X-2 obtained over 4 yr, which gives an improved mass function of 0.69 +/- 0.03 Msolar (1 sigma ). In addition, we resolve the rotationally broadened absorption features of the secondary star for the first time, deriving a rotation speed of v{sin}i=34.2+/-2.5 km s-1 (1 sigma ), which leads to a mass ratio of q=Mc/M_{{X}}=0.34 +/- 0.04 (1 sigma , assuming a tidally locked and Roche lobe-filling secondary). Hence, with the lack of X-ray eclipses (i.e., i<~73^° ) we can set firm 95% confidence lower limits to the neutron star mass of M_{{X}}>1.27 Msolar and to the companion star mass of Mc>0.39 Msolar. However, by additionally requiring that the companion must exceed 0.75 Msolar (as required theoretically to produce a steady low-mass X-ray binary), then M_{{X}}>1.88 Msolar and i<61^° (95% confidence lower and upper limit, respectively), thereby making Cyg X-2 the highest mass neutron star measured to date. If confirmed, this would set significant constraints on the equation of state of nuclear matter.