Spatial division multiplexing in multi-mode fibers allows to largely enhance transmission capacity compared to single-mode links. Photonic integrated circuits can provide solutions for mode multiplexing at the transmitter and demultiplexing at the receiver but have to generally face high losses and inter-modal cross-talk issues. Here a photonic circuit for efficient mode multiplexing and demultiplexing in few-mode fibers is presented and demonstrated. Two 10 Gbit/s channels at the same wavelength and polarization are simultaneously transmitted over modes LP01 and LP11a of a few-mode fiber relying only on integrated mode MUX and DEMUX. The proposed Indium-Phosphide-based circuits have a good coupling efficiency with fiber modes and mode-dependant loss smaller than 1 dB. Measured mode excitation cross-talk is as low as -20 dB and a channel cross-talk after propagation and demultiplexing of -15 dB is achieved. An operational bandwidth of the full transmission system of at least 10 nm is demonstrated. Both mode MUX and DEMUX are fully reconfigurable and allow a dynamic switch of channel routing in the transmission system. These results enable fully-integrated fiber mode handling for high-bandwidth flexible optical networks.