Private company must have minimum two directors and hence appointment of one more director is mandatory in the given case. The only issue is when to appointment and who will appoint.
Legally speaking, Sec.174(2) provides that the continuing director may act for the purpose of increasing the number of directors to that fixed for the quorum, or of summoning general meeting of the company and for no other purpose. Hence remaining one director can also appoint new director. However, practically speaking the system may not accept DIR-12 for death of director and may give message that the number of directors has reduced below minimum fixed by the Act. Hence it will be advisable to appoint someone as new director on or before the date of death and file DIR-12 accordingly.