The question itself seems to be hazy. Let the questioner clarify the following first:
Whether the length of the notice period served by the employee is the same as mentioned in the contract of employment?
Whether there was formal acceptance of the resignation before the end of the notice period?
If accepted how the employee continued to be in service even after the expiry of the notice period?
Whether the resignation was rejected at the end of the notice period on some valid grounds like the pendency or contemplation of any disciplinary action against him? (if so, he has to continue his services).
In such situation, if the one month period falls after the finality of the disciplinary proceedings i.e., with award of punishment or exoneration or dropping of charges, he has to submit his resignation afresh complying with the notice provision once again.
If the employer simply fails to take any action on the earlier resignation submitted by the employee even after the expiry of notice period as provided for in the contract of employment, the employee has to make a formal request once again to accept the resignation and relieve him forthwith as he has already served the notice period. Legally speaking, on such an occasion the employer has to relieve the employee formally without insisting on a fresh notice.