**Abstract** : Context. Two years ago, the Ams-02 collaboration released the most precise measurement of the cosmic ray positron flux. In the conventional approach, in which positrons are considered as purely secondary particles, the theoretical predictions fall way below the data above 10 GeV. One suggested explanation for this anomaly is the annihilation of dark matter particles, the so-called weakly interactive massive particles (WIMPs), into standard model particles. Most analyses have focused on the high-energy part of the positron spectrum, where the anomaly lies, disregarding the complicated GeV low-energy region where Galactic cosmic ray transport is more difficult to model and solar modulation comes into play.Aims. Given the high quality of the latest measurements by Ams-02, it is now possible to systematically re-examine the positron anomaly over the entire energy range, this time taking into account transport processes so far neglected, such as Galactic convection or diffusive re-acceleration. These might impact somewhat on the high-energy positron flux so that a complete and systematic estimate of the secondary component must be performed and compared to the Ams-02 measurements. The flux yielded by WIMPs also needs to be re-calculated more accurately to explore how dark matter might source the positron excess. Methods. We devise a new semi-analytical method to take into account transport processes thus far neglected, but important below a few GeV. It is essentially based on the pinching of inverse Compton and synchrotron energy losses from the magnetic halo, where they take place, inside the Galactic disc. The corresponding energy loss rate is artificially enhanced by the so-called pinching factor, which needs to be calculated at each energy. We have checked that this approach reproduces the results of the Green function method at the per mille level. This new tool is fast and allows one to carry out extensive scans over the cosmic ray propagation parameters.Results. We derive the positron flux from sub-GeV to TeV energies for both gas spallation and dark matter annihilation. We carry out a scan over the cosmic ray propagation parameters, which we strongly constrain by requiring that the secondary component does not overshoot the Ams-02 measurements. We find that only models with large diffusion coefficients are selected by this test. We then add to the secondary component the positron flux yielded by dark matter annihilation. We carry out a scan over WIMP mass to fit the annihilation cross-section and branching ratios, successively exploring the cases of a typical beyond-the-standard-model WIMP and an annihilation through light mediators. In the former case, the best fit yields a p-value of 0.4% for a WIMP mass of 264 GeV, a value that does not allow to reproduce the highest energy data points. If we require the mass to be larger than 500 GeV, the best-fit χ2 per degree of freedom always exceeds a value of 3. The case of light mediators is even worse, with a best-fit χ2 per degree of freedom always larger than 15.Conclusions. We explicitly show that the cosmic ray positron flux is a powerful and independent probe of Galactic cosmic ray propagation. It should be used as a complementary observable to other tracers such as the boron-to-carbon ratio. This analysis shows also that the pure dark matter interpretation of the positron excess is strongly disfavoured. This conclusion is based solely on the positron data, and no other observation, such as the antiproton flux or the CMB anisotropies, needs to be invoked.