  The HEGRA system of 4 Imaging Atmospheric Cherenkov Telescopes (IACTs) has
been used to determine the flux and the spectrum of cosmic ray protons over a
limited energy range around 1.5 TeV. Although the IACT system is designed for
the detection of gamma-rays with energies above 500 GeV, it has also a large
detection area of $ \simeq 10^6 m^2 \cdot 3 msr$ for primary protons of
energies above 1 TeV and the capability to reconstruct the primary proton
energy with a reasonable accuracy $\Delta$E/E of 50% near this threshold.
Furthermore, the principle of stereoscopic detection of air showers permits the
effective suppression of air showers induced by heavier primaries already on
the trigger level, and in addition on the software level by analysis of the
stereoscopic images. The combination of both capabilities permits a
determination of the proton spectrum almost independently of the cosmic ray
chemical composition. The accuracy of our estimate of the spectral index at 1.5
TeV is limited by systematic uncertainties and is comparable to the accuracy
achieved with recent balloon and space borne experiments. In this paper we
describe in detail the analysis tools, namely the detailed Monte Carlo
simulation, the analysis procedure and the results. We determine the local
(i.e. in the range of 1.5 to 3 TeV) differential spectral index to be $\gamma_p
= 2.72\pm 0.02_{stat.}{\pm 0.15}_{syst.}$ and obtain an integral flux above 1.5
TeV of $F(> 1.5 TeV) =3.1\pm 0.6_{stat.}\pm 1.2_{syst.} 10^{-2}/s sr m^2$.
