We study the thermopower of a disordered nanowire in the presence of an external gate electrode which can be used for depleting the carrier density inside the nanowire. In this first paper, we study the low temperature regime where the electron transport remains elastic through a disordered nanowire described by a one-dimensional tight-binding Anderson model. The gate voltage is depicted by a uniform on-site potential which can be varied for shifting the impurity band of the nanowire. The thermopower is evaluated using a Sommerfeld expansion valid below a characteristic temperature which depends on the gate potential. In the limit where the length of the nanowire exceeds the localization length, the typical thermopower is given by three analytical expressions describing the cases where the electron transport takes place (i) inside the impurity band of the nanowire, (ii) around its band edges and eventually (iii) outside its band. Notably we highlight that the typical thermopower is strongly enhanced at the band edges, an analytical formula obtained by Derrida and Gardner for describing the energy dependence of the localization length around the band edges allowing us to perfectly describe the edge behavior of the thermopower. Then, we investigate the mesoscopic fluctuations of the thermopower around its typical value. Inside the impurity band, they are given by a Lorentzian distribution, with a width proportional to the density of states evaluated at the Fermi energy. As the band edges are approached, we find a transition towards a Gaussian distribution. Eventually, the implications of our results for the low temperature figure of merit and the output power of disordered nanowires are discussed.