The first measurement of dielectron ($\mathrm{e}^{+}\mathrm{e}^{-}$) production in central (0$-$10$\%$) Pb--Pb collisions at a center-of-mass energy per nucleon pair $\mathbf{\sqrt{{\textit{s}}_{\mathrm{NN}}}}$ = 2.76 TeV at the LHC is presented. The dielectron invariant-mass spectrum is measured in the ALICE central barrel detectors for electrons with pseudorapidity $|\eta_{\mathrm{e}}|<0.8$ and transverse momentum $0.4 < p_{\mathrm{T}, \mathrm{e}} < 5$ GeV/$\textit{c}$. The dielectron invariant-mass spectrum is compared to the expected contributions from known hadron decays in the invariant-mass range $0 < m_{\mathrm{ee}}<3.5\ \mathrm{GeV}/\textit{c}^{2}$. The ratio of data and the cocktail of hadronic contributions without vacuum $\rho^...