  As is well-known, in Bogoliubov's theory of an interacting Bose gas the
ground state of the Hamiltonian $\hat{H}=\sum_{\bf k\neq 0}\hat{H}_{\bf k}$ is
found by diagonalizing each of the Hamiltonians $\hat{H}_{\bf k}$ corresponding
to a given momentum mode ${\bf k}$ independently of the Hamiltonians
$\hat{H}_{\bf k'(\neq k)}$ of the remaining modes. We argue that this way of
diagonalizing $\hat{H}$ may not be adequate, since the Hilbert spaces where the
single-mode Hamiltonians $\hat{H}_{\bf k}$ are diagonalized are not disjoint,
but have the ${\bf k}=0$ in common. A number-conserving generalization of
Bogoliubov's method is presented where the total Hamiltonian $\hat{H}$ is
diagonalized directly. When this is done, the spectrum of excitations changes
from a gapless one, as predicted by Bogoliubov's method, to one which has a
finite gap in the $k\to 0$ limit.
