Context. Past observations of Saturn with ground-based and space telescopes have enabled the monitoring of tropospheric wind speeds using cloud-tracking techniques. The most remarkable feature is a broad and fast prograde jet at the equator that reaches speeds of ∼400 m/s. Saturn's stratospheric dynamics are less well-known. At low latitudes, they are characterized by the thermal signature of an equatorial oscillation; the observed thermal structure implies that there is a strong oscillating vertical shear of the zonal winds throughout the stratosphere. However, wind speeds in this region cannot be measured by cloud-tracking techniques and remain unknown. Aims. The objective of this study is to measure directly and for the first time the zonal winds in Saturn's stratosphere using the ALMA interferometer. Methods. We observed the spectral lines of CO at 345.796 GHz and HCN at 354.505 GHz with the high spatial (∼0.6 ′′ ×0.5 ′′) and spectral resolutions enabled by ALMA, and measured the Doppler shift induced by the winds on the lines at the planet limb where the emission is the strongest. After subtracting the beam-convolved planet rotation, we derived the zonal wind speeds as a function of latitude. Results. We measured the zonal winds from ∼20 • S to the northern polar latitudes. Latitudes between 20 • S and 45 • S were obscured by the rings and were inaccessible southward of 45 • S. The zonal wind profiles obtained on the eastern and western limbs are consistent within the error bars and probe from the 0.01 to the 20 mbar level. We most noticeably detect a broad super-rotating prograde jet that spreads from 20 • S to 25 • N with an average speed of 290±30 m/s. This jet is asymmetrical with respect to the equator, a possible seasonal effect. We tentatively detect the signature of the Saturn semi-annual oscillation (SSAO) at the equator, in the form of a ∼-50±30 m/s peak at the equator which lies on top of the super-rotating jet. We also detect a broad retrograde wind (-45±20 m/s) of about 50 m/s in the mid-northern latitudes. Finally, in the northern polar latitudes, we observe a possible auroral effect in the form of a ∼200 m/s jet localized on the average position of the northern main auroral oval and in couter-rotation, like the Jovian auroral jets. Conclusions. Repeated observations are now required to monitor the temporal evolution of the winds and quantify the variability of the SSAO jet, to test the seasonality of the asymmetry observed in the broad super-rotating jet, and to verify the presence of auroral jets in the southern polar region of Saturn.