Five games of the ban are suspended for two years and three have already been served - the second Test against New Zealand and two T20 Blast matches for Sussex Sharks.
The punishment was made by the Cricket Discipline Commission (CDC), an independent panel that adjudicated on the England and Wales Cricket Board's (ECB) rules and also fined Robinson £3,200.
"I fully accept the CDC's decision," Robinson told BBC Sport. "As I have said previously, I am incredibly embarrassed and ashamed about the tweets I posted many years ago and apologise unreservedly for their contents.
"I am deeply sorry for the hurt I caused to anyone who read those tweets and in particular to those people to whom the messages caused offence. This has been the most difficult time in my professional career for both my family and myself.
"Whilst I want to move on, I do want to use my experience to help others in the future through working with the PCA [player's body the Professional Cricketers' Association]."
Following his suspension for the second Test against New Zealand, Robinson took a short break but he returned to action towards the end of June when he played for Sussex Sharks in a T20 Blast game against Gloucestershire.
Ironically he has since had to miss matches because of having to isolate after Sussex team-mate Tom Clark tested positive for Covid.
Robinson will be clear to play any day now, though, and will be hoping for a call-up by England for the forthcoming Test seires with India - after starring with bat and ball in his only Test match to date.
The CDC said it took into account "the nature and content of the tweets, the breadth of their discrimination, their widespread dissemination in the media and the magnitude of the audience to whom they became available". It added it also considered "significant mitigation, including the time that had elapsed since the tweets were posted" by Robinson when he was aged 18 and 19.
Robinson chose to address the panel, which found him to be "a very different person to the one who sent the tweets" and the CDC "also took account of his remorse, admissions and cooperation as well as the huge impact which the revelation of these tweets and its consequences have had upon him and his family".
"Ollie has acknowledged that, whilst published a long time ago when he was a young man, these historic tweets were unacceptable," said ECB chief executive Tom Harrison.
"He has engaged fully in the disciplinary process, admitted the charges, has received his sanction from the CDC and will participate in training and use his experiences to help others. We stand against discrimination of all forms, and will continue working to ensure cricket is a welcoming and inclusive sport for all."