The European Galileo system offers one dedicated signal that is superior to all other signals currently available in space, namely the broadband signal E5. This signal has a bandwidth of at least 51 MHz using an AltBOC modulation. It features a code range noise at centimeter level. Additionally, the impact of multipath effects on this signal is significantly lower compared to all other available GNSS signals. These unique features of Galileo E5 drastically improve the precision of code range measurements and hence enable precise single-frequency positioning. Certain scientific and non-scientific applications in the positioning domain could likely benefit from the exploitation of E5 measurements. A positioning approach based on an additive combination of code range and carrier phase measurements (CPC--"code-plus-carrier") to eliminate the ionospheric delay could be used to perform precise positioning over long distances. Unfortunately, this derived observable contains the ambiguity term as an additional unknown what normally requires longer observation windows in order to allow sufficient convergence of the ambiguity parameters. For this reason, a rapid convergence algorithm based on Kalman filtering was implemented in addition to the conventional CPC approach that is also discussed. The CPC-based results yield a positioning precision of 2---5 cm after a convergence time of about 3 h. The rapid convergence filter allows fixing the ambiguity terms within a few minutes, and the obtained position results are at the sub-decimeter level. Regarding one selected test, real data from Galileo experimental satellite GIOVE A were used in order to confirm our assumptions. However, since the current Galileo constellation is not sufficient for real-world positioning trials yet, all major results are based on simulated data.