We calculate the neutrino anapole moment in the context of the standard model of the electroweak interactions and we show that this quantity is infinite and gauge dependent in this model. We define the electroweak anapole moment of the neutrino ${a}_{{\ensuremath{\nu}}_{l}}^{\mathrm{EW}}$ through the ${\ensuremath{\nu}}_{l}{l}^{\ensuremath{'}}$ scattering at the one-loop level. This amplitude can be written, in a certain kinematical region, in terms of only two form factors, which multiply the electromagnetic current and the weak neutral current, respectively. By construction, ${a}_{{\ensuremath{\nu}}_{l}}^{\mathrm{EW}}$ is defined in such a way that it becomes a finite and gauge independent quantity, which gets contributions from the proper neutrino electromagnetic vertex, from the $\ensuremath{\gamma}\ensuremath{-}Z$ self-energy, and from part of the box diagrams. We obtain that ${a}_{{\ensuremath{\nu}}_{l}}^{\mathrm{EW}}$ is of the order ${10}^{\ensuremath{-}34} {\mathrm{cm}}^{2}$