A 5.0-at.% Nd-doped La0.64Gd0.41Sc2.95(BO3)4 (Nd:LGSB) borate laser crystal was successfully grown by the Czochralski method, for the first time to our knowledge. The spectroscopic properties of the grown crystal are discussed and 1 µm laser emission, under end-pumping with a fiber-coupled diode laser at 807 nm, is reported. A c-cut Nd:LGSB medium yielded 1.35 W continuous-wave output power at 0.63 overall optical-to-optical efficiency, with respect to the absorbed pump power, together with the high 0.68 slope efficiency. With an a-cut Nd:LGSB sample, 0.81 W output power at 0.52 optical-to-optical efficiency was obtained. The laser emission performances under quasi-continuous wave pumping are presented as well, for both c-cut and a-cut crystals. Passive Q-switching was investigated with a semiconductor saturable absorber mirror (SESAM). Laser pulses with 2.2 µJ energy and 32.8 ns durations were recorded from a-cut Nd:LGSB. The average output power reached 0.36 W at 1.55 W absorbed pump power. Passive mode-locking with SESAM was achieved in a long Z-type resonator. Ultrashort pulses with 0.19 W average power, 1.63 nJ energy, and 1.43 ps pulse duration, at 118 MHz repetition rate, are demonstrated for the a-cut Nd:LGSB medium.