A process for producing sugar from beets includes the step of filtering a sucrose-containing feed juice, which has been obtained by diffusion from sliced sugar beets, through a first ultrafiltration membrane that has a first molecular weight cutoff. This ultrafiltration step produces a first ultrafiltration permeate and a first ultrafiltration retentate. The first ultrafiltration permeate is filtered through a second ultrafiltration membrane that has a second molecular weight cutoff that is lower than the first molecular weight cutoff. This second ultrafiltration step produces a second ultrafiltration permeate and a second ultrafiltration retentate. The second ultrafiltration permeate is nanofiltered through a nanofiltration membrane, thereby producing a nanofiltration permeate and a nanofiltration retentate. The nanofiltration retentate has a higher concentration of sucrose on a dry solids basis than the feed juice in step (a), and can be used in evaporation and crystallization operations to produce crystals of white sugar. The process can optionally include ion exchange and/or electrodialysis purification steps, prior to or after the nanofiltration step. Recycle syrups can be treated with enzyme or a chromatographic separator to remove raffinose.