The 26 year old appeared at South East Northumberland Magistrates' Court dressed in a black T-shirt and jeans to plead guilty to both offences.
The court heard that, on November 5 and November 20 last year, a car owned by the France international was caught speeding in Gosforth, Newcastle.
As a result Ben Arfa was notified by police but the magistrates were told he failed to respond at all to one of the incidents and waited two months before responding to the other allegation.
Neil Fodor, prosecuting, said that, as the registered keeper of the car, he had an obligation to respond in the specified time.
"Whatever situation, whatever procedures Mr Ben Arfa had in place for being abroad and where English was not his first language, it was his responsibility that those procedures worked and clearly on these occasions they did not," he said.
Michelle Brown, defending, said Ben Arfa had been going back and forth from the UK and France at the time.
"Being away from the home address, the system where the letters are sent on did not go through the necessary chain with the quickness that was required," she said.
"I hope the court accepts his genuine apology."
Sentencing Ben Arfa to a six-month driving ban, 12 points on his licence and a total fine of £2,680, chairman of the bench Margaret Sewell said if he was to break the ban he could face imprisonment.