  We propose an algorithm for the application of the Laplace method for the
calculation of the simplest Feynman diagram with a single loop in the scalar
{\phi}^3 theory. The calculation of the contribution of such a diagram to the
scattering amplitude requires the calculation of a fourfold integral over the
four-momenta components circulating in a loop. The essence of the Laplace
method for the calculation of multiple integrals lies in the fact that if the
module of an integrand has a point of sufficiently sharp maximum inside the
integration domain, then the integral can be replaced by a Gaussian integral by
representing the integrand in the form of an exponent from the logarithm and
expanding this logarithm into Taylor series in the vicinity of a maximum point
up to the second degree terms. We show that there are two-dimensional and
non-intersecting surfaces inside the four-dimensional region of integration, on
which the maximum of the module of integrand is reached. This leads to a
problem that the integrand is non-analytically dependent on the parameters
responsible for bypassing the poles. Also the derivatives of logarithm of the
scattering amplitude are non-analytically dependent on these parameters.
However, in the paper we show that these non-analyticities compensate each
other. As a result of such a procedure, three of the four integrations can be
done analytically, and the calculation of the contribution of the diagram to
the scattering amplitude is reduced to a numerical calculation of the single
integral in finite bounds from an expression that does not contain
non-analyticities. The described calculation method is used to construct a
model dependence of elastic scattering differential cross section
d{\sigma}_elastic/dt on the square of the transmitted four-momentum t
(Mandelstam variable).
