We define a complete sample of thirty-three GHz-Peaked-Spectrum (GPS) radio sources based on their spectral properties. We present measurements of the radio spectra and polarization of the complete sample and a list of additional GPS sources which fail one or more criteria to be included in the complete sample. The majority of the data have been obtained from quasi-simultaneous multi-frequency observations at the Very Large Array (VLA) during 3 observing sessions. Low frequency data from the Westerbork Synthesis Radio Telescope (WSRT) and from the literature have been combined with the VLA data in order to better define the spectral shape. The objects presented here show a rather wide range of spectral indices at high and low frequencies, including a few cases where the spectral index below the turnover is close to the theoretical value of 2.5 typical of self-absorbed incoherent synchrotron emission. Faint and diffuse extended emission is found in about 10% of the sources. In the majority of the GPS sources, the fractional polarization is found to be very low, consistent with the residual instrumental polarization of 0.3%.