  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).
