When both two-electron textit{and} two-hole Cooper-pairing are treated on an equal footing in the ladder approximation to the Bethe-Salpeter (BS) equation, the zero-total-momentum Cooper-pair energy is found to have two textit{real} solutions $mathcal{E}_{0}^{BS}=pm 2hbar omega_{{D}%}/sqrt{{e}^{2/lambda }+{1}}$ which coincide with the zero-temperature BCS energy gap $Delta =hbar omega_{D}/sinh (1/lambda) $ in the weak coupling limit. Here, $hbar omega_{D}$ is the Debye energy and $lambda geq 0$ the BCS model interaction coupling parameter. The interpretation of the BCS energy gap as the binding energy of a Cooper-pair is often claimed in the literature but, to our knowledge, never substantiated even in weak-coupling as we find here. In addition, we confirm the two purely-textit{imaginary} solutions assumed since at least the late 1950s as the textit{only} solutions, namely, $mathcal{E}_{0}^{BS}=pm i2hbar omega_{{D}}/sqrt{{e}^{2/lambda}{-1}}.$
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