In early November 2017, the first two nonexperimental third-generation spacecraft of China’s indigenous BeiDou Navigation Satellite System (BDS) were placed in medium Earth orbit (MEO) at an altitude of approximately 21,500 km. As of today, the BDS constellation includes ten of these MEO vehicles. Although reportedly not yet in operational service, the satellites continuously transmit a legacy B1 open service signal on a frequency of 1561.098 MHz, which is being tracked by a global set of approximately 50 ground receivers of the International GNSS Service (IGS) station network. In this work, we report on initial efforts at the European Space Agency’s IGS analysis center to generate the first orbit and clock solutions for the new BeiDou-3 series of satellites using 24-hour nonoverlapping processing arcs and B1 single-frequency data from the IGS tracking network. First-order ionospheric delays are removed by averaging the B1 phase and pseudorange observables. Data post-fit residuals have a root mean square (RMS) value of less than 0.2 m and reveal—unlike the pseudoranges in BeiDou-2 processing—few or no systematic signatures. Comparisons of adjacent orbital arcs at the midnight epoch indicate 3D orbit accuracies better than 1 m (1-sigma). The radial orbit component is accurate to better than 0.2 m (1-sigma) according to initial satellite laser ranging (SLR) data. RMS differences relative to the predicted ephemeris from the BDS broadcast navigation message are at the level of 1 m for the satellite orbits and 3 ns for the clocks, respectively. We finally address satellite-to-satellite differences in the direct radiation pressure parameter estimates of about 70 nm/s2, suggesting that only seven of ten MEOs have completely unfolded their solar arrays yet, as well as differences in the estimates of satellite antenna offsets, indicating different locations of the L-band navigation antenna relative to the spacecraft’s center of gravity.