The present disclosure discloses a process for accelerating, increasing, and stabilizing production of biogas with a high methane content in systems for biodigestion of organic waste. The process comprises: a) obtaining from an anaerobic biodigester (3) a fraction of organic waste equal in weight to K times the weight of the daily organic waste load of the anaerobic biodigester (3), wherein the parameter K is a numeric value selected between 10−3 and 10−1, sending the fraction to one or more acceleration devices (4), retaining the fraction in the one or more acceleration devices (4) until a final concentration (Cf) of the methanogenic bacteria in the fraction is equal to M times an initial concentration (Ci), wherein the parameter M is a numeric value selected between 103 and 108; b) as soon as said final concentration (Cf) is obtained in the fraction, directing said fraction from the acceleration device (4) back to the anaerobic biodigester (3); and c) successively repeating steps a) and b).