Cryne announced in September that he intended to step aside after being diagnosed with terminal cancer, opening the door for new investment streams.
Investors Paul Conway, Grace Hung and Neerav Parekh have been added to the board as part of the takeover, meanwhile, which chief executive Gauthier Ganaye hopes will signal "an exciting chapter" for Barnsley.
"We would like to thank every member of our loyal fan base for their patience and continued support over recent months," Ganaye told the club's official website.
"An exciting chapter is opening for Barnsley Football Club and we are looking forward to the future together."
Barnsley are 20th in the Championship, four points above the relegation zone.