Lauber last week offered to resign after a court said he covered up the meeting and lied to supervisors during an investigation by his office into corruption surrounding Fifa.
Both have denied any wrongdoing.
Lauber's office said he officially tendered his resignation on Tuesday, with his last day of active duty set for 31 August.
It comes after special prosecutor Stefan Keller was appointed last month to review criminal complaints against the two men and others.
He found indications of criminal conduct related to their meetings, the authority overseeing the Office of the Attorney General (OAG) said.
"This concerns abuse of public office, breach of official secrecy, assisting offenders and incitement to these acts," the watchdog overseeing the OAG said in a statement.
The OAG said it had taken note of the development, adding Lauber would make statements to parliamentary committees if necessary.
The watchdog said Keller had now opened proceedings against both Infantino and a regional public prosecutor who was involved in the meetings, and was seeking parliamentary approval to have Lauber's immunity from prosecution waived.
Fifa is based in the Swiss city of Zurich. BBC Sport has contacted the governing body for a response.