Aims. We present a new measurement of the cosmic X-ray background (CXRB) in the 1.5-7 keV energy band, performed by exploiting the Swift X-ray telescope (XRT) data archive. We also present a CXRB spectral model in a wider energy band (1.5-200 keV), obtained by combining these data with the recently published Swift-BAT measurement. Methods. From the XRT archive we collect a complete sample of 126 high Galactic latitude gamma-ray burst (GRB) follow-up observations. This provides a total exposure of 7.5 Ms and a sky-coverage of ~7 square degrees which represents a serendipitous survey, well suited for a direct measurement of the CXRB in the 1.5-10 keV interval. Our work is based on a complete characterization of the instrumental background and an accurate measurement of the stray-light contamination and vignetting calibration. Results. We find that the CXRB spectrum in the 1.5-7 keV energy band can be equally well fitted by a single power-law with photon index $\Gamma=1.47\pm0.07$ or a single power-law with photon index $\Gamma=1.41\pm0.06$ and an exponential roll-off at 41 keV. The measured flux in the 2-10 keV energy band is $2.18 \pm0.13 \times10^{-11}$ erg cm-2 s-1 deg-2 in the 2-10 keV band. Combining Swift-XRT with Swift-BAT (15-200 keV) we find that, in the 1.5-200 keV band, the CXRB spectrum can be well described by two smoothly-joined power laws with the energy break at $29.0\pm0.5$ keV corresponding to a $\nu F_{\nu}$ peak located at $22.4\pm0.4$ keV. Conclusions. Taking advantage of both the Swift high energy instruments (XRT and BAT), we produce an analytical description of the CXRB spectrum over a wide (1.5-200 keV) energy band. This model is marginally consistent with the HEAO1 measurement (~10% higher) at energies higher than 20 keV, while it is significantly (30%) higher at low energies (2-10 keV).