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We present a measurement of the polarization-temperature angular cross power spectra,⟨ TE⟩ and⟨ TB⟩, of the cosmic microwave background. The result is based on~ 200 hr of data from eight polarization-sensitive bolometers operating at 145 GHz during the 2003 flight of BOOMERANG. We detect a significant⟨ TE⟩ correlation in the l-range between 50 and 950 with a statistical significance of> 3.5 σ. Contamination by polarized foreground emission and systematic effects are negligible in comparison with statistical uncertainties. The spectrum is consistent with previous detections and with the" concordance model" that assumes adiabatic initial conditions. This is the first measurement of polarization-temperature angular cross-power spectra using bolometric detectors.