  The TeV and X-ray data obtained by the imaging Cherenkov telescope CAT and
X-ray satellite BeppoSAX during the remarkable flare of Mkn 501 in April 16,
1997 are used to constrain the flux of the Cosmic Infrared Background (CIB)
using different CIB models. We show that a non-negligible absorption of
gamma-rays due to the CIB could take place already in the low-energy (sub-TeV)
domain of the spectrum of Mkn 501. This implies that the data of the low-energy
threshold CAT telescope contain very important information about the CIB at
short wavelengths, 0.4 mum <= lambda <= 3. mum. The analysis of almost
simultaneous spectroscopic measurements of Mkn 501 in a high state by CAT and
BeppoSAX in the framework of the standard homogeneous Synchrotron-Self-Compton
(SSC) framework model leads to the conclusion that the density of the
near-infrared background with typical ``starlight spectrum'' around 1 mum
should be between 5 and 35 nW m^-2 sr^-1 (99 % CL), with most likely value
around 20 nW m^-2 sr^-1. Also we argue that the CAT gamma-ray data alone allow
rather robust upper limits on the CIB, lambda F_lambda <= 60 nW m^-2 sr^-1 at 1
mum, taking into account that for any reasonable scenario of gamma-ray
production the differential intrinsic spectrum of gamma-ray hardly could be
flatter than dN/dE == E^-1. This estimate agrees, within statistical and
systematic uncertainties, with recent reports about tentative detections of the
CIB at 2.2 and 3.5 mum by the Diffuse Infrared Background Experiment (DIRBE),
as well as with the measurements of the background radiation at optical
wavelengths from absolute photometry. We also discuss the impact of the
intergalactic absorption effect in derivation of the SSC parameters for the jet
in Mkn 501.
