About 50 years ago, Sarachick and Schappert [Phys. Rev. D. 1, 2738-2752 (1970)] showed that relativistic Thomson scattering leads to wavelength shifts that are proportional to the laser intensity. About 28 years later, Chen et al. [Nature 396, 653-655 (1998)] used these shifts to estimate their laser intensity near 10(18) W/cm(2). More recently, there have been several theoretical studies aimed at exploiting nonlinear Thomson scattering as a tool for direct measurement of intensities well into the relativistic regime. We present the first quantitative study of this approach for intensities between 10(18) and 10(19) W/cm(2). We show that the spectral shifts are in reasonable agreement with estimates of the peak intensity extracted from images of the focal area obtained at reduced power. Finally, we discuss the viability of the approach, its range of usefulness and how it might be extended to gauge intensities well in excess of 10(19) W/cm(2). (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement