Limits on the Polarization of the Cosmic Microwave Background Radiation at Multipoles up to l~2000

We report upper limits on the polarization of the CMBR as measured with the Cosmic Background Imager, a 13 element interferometer that operates in the 26-36 GHz band and is sited on Llano de Chajnantor in northern Chile. The array consists of 90-cm Cassegrain antennas mounted on a steerable platform that can be rotated about the optical axis to facilitate polarization observations. The CBI employs single-mode circularly polarized receivers and it samples multipoles from l~400 to l~3500. The polarization data were calibrated on 3C279 and Tau A. The polarization observations consist of 278 hours of data on two fields taken in 2000, during the first CBI observing season. A joint likelihood analysis of the two fields yields three upper limits (95% c.l.) for \mathcal{C}_l^{EE} = C^{EE}l(l+1)/(2pi) under the assumption that \mathcal{C}_l^{BB}=0: 49.0 microK^2 (l=603); 164 microK^2 (l=1144); and 630 microK^2 (l=2048).