  We formulate a conserving gapless mean-field theory for Bose-Einstein
condensates on the basis of a Luttinger-Ward thermodynamic functional. It is
applied to a weakly interacting uniform gas with density $n$ and s-wave
scattering length $a$ to clarify its fundamental thermodynamic properties. It
is found that the condensation here occurs as a first-order transition. The
shift of the transition temperature $\Delta T_c$ from the ideal-gas result
$T_{0}$ is positive and given to the leading order by $\Delta T_c = 2.33a
n^{1/3}T_0$, in agreement with a couple of previous estimates. The theory is
expected to form a new theoretical basis for trapped Bose-Einstein condensates
at finite temperatures.
