Single-longitudinal-mode light sources on a silicon platform exhibit tremendous potential in data transmission, integrated on-chip optical interconnects, and optical ranging. A unique InAs/GaAs quantum dot discrete mode (DM) laser directly grown on a silicon-on-insulator (SOI) substrate with embedded silicon gratings has been demonstrated here, which is regrowth-free and compatible with integrated silicon photonics. Attributed to refractive index perturbation and optical feedback from embedded silicon gratings prepatterned on the SOI substrate, this DM laser can provide stable single-mode operation with a threshold current of 45 mA and an output power of 9 mW. The maximum optical signal-to-noise ratio (OSNR) achieved is 43.6 dB with an optical linewidth of 157 kHz. The 3 dB direct modulation bandwidth is measured as 3.6 GHz. The DM laser can be externally modulated at 70 Gbit/s with non-return-to-zero (NRZ) modulation format and 80 Gbit/s under 4-level pulsed amplitude (PAM4) modulation format. Such SOI-based DM laser paves the way toward large-scale and high-power on-chip integrated lasers.