  We present the results of the analysis of the broad-band spectrum of Cygnus
X-1 from 3.0 to 200 keV, using data from a 10 ksec observation by the Rossi
X-ray Timing Explorer. The spectrum can be well described phenomenologically by
an exponentially cut-off power law with a photon index Gamma = 1.45 +/- 0.02 (a
value considerably harder than typically found), e-folding energy E_fold = 162
+/- 9 keV, plus a deviation from a power law that formally can be modeled as a
thermal blackbody with temperature kT_bb = 1.2 +/1 0.2 keV. Although the 3 - 30
keV portion of the spectrum can be fit with a reflected power law with Gamma =
1.81 +/- 0.01 and covering fraction f = 0.35 +/- 0.02, the quality of the fit
is significantly reduced when the HEXTE data in the 30 - 100 keV range is
included, as there is no observed hardening in the power law within this energy
range. As a physical description of this system, we apply the accretion disc
corona models of Dove, Wilms & Begelman (1997) --- where the temperature of the
corona is determined self-consistently. A spherical corona with a total optical
depth tau = 1.6 +/- 0.1 and an average temperature kT_c = 87 +/- 5 keV,
surrounded by an exterior cold disc, does provide a good description of the
data (reduced chi-squared = 1.55). These models deviate from the data by up to
7% in the 5 - 10 keV range, and we discuss possible reasons for these
discrepancies. However, considering how successfully the spherical corona
reproduces the 10 - 200 keV data, such ``photon-starved'' coronal geometries
seem very promising for explaining the accretion processes of Cygnus X-1.
