  Assuming that supernovae type Ia (SNe Ia) are standard candles one could use
them to test cosmological theories. The Hubble Space Telescope team analyzed
186 SNe Ia\cite{Riess_04} to test the Standard Cosmological model (SC)
associated with expanded lengths in the Universe and evaluate its parameters.
We use the same sample to determine parameters of Conformal Cosmological model
(CC) with relative reference units of intervals, so that conformal quantities
of General Relativity are interpreted as observables. We concluded, that really
the test is extremely useful and allows to evaluate parameters of the model.
From a formal statistical point of view the best fit of the CC model is almost
the same quality approximation as the best fit of SC model with
$\Omega_\Lambda=0.72, \Omega_m=0.28$. As it was noted earlier, for CC models, a
rigid matter component could substitute the $\Lambda$-term (or quintessence)
existing in the SC model. We note that a free massless scalar field can
generate such a rigid matter. We describe results of our analysis for more
recent "gold" data (for 192 SNe Ia).
