  We discuss mechanisms of decoherence of a confined Bose-Einstein condensate
at finite temperatures under the explicit condition of conservation of the
total number of bosons $N$ in the trap. A criterion for the irreversible decay
of the condensate two-time correlator is formulated in terms of the {\it
Orthogonality Catastrophe} (OC) for the exact N-body eigenstates, so that no
irreversible decay occurs without the OC. If an infinite external bath contacts
a finite condensate, the OC should practically always occur as long as the bath
degrees o freedom are interacting with each other. We claim that, if no
external bath is present and the role of the bath is played by the normal
component, no irreversible decay occurs. We discuss the role of the effect of
the {\it level repulsion} in eliminating the OC. At finite temperatures, the
time-correlations of the condensate isolated from the environment are dominated
by the reversible dephasing which results from the thermal ensemble averaging
over realizations of the normal component. Accordingly, the correlator exhibits
the gaussian decay with certain decay time $\tau_d$ dependent on temperature as
well as on intensity of the shot noise determined by the statistical
uncertainty in the number of bosons $N$ deposited into the trap. We estimate
$\tau_d$ for the conditions resembling those realized in the JILA experiment,
and obtain the value close to 1s, if no shot noise is taken into account. At
temperatures as large as 0.5 of the temperature of the Bose-Einstein
condensation, $\tau_d \leq 100$ms. Depending on the variance $\Delta N$ of $N$,
inclusion of the shot noise may shorten these times.
