I would have got the answer wrong if I had not checked in the rules as I thought you would continue from where black and white are. In fact it seems that the black should be re-spotted.
This is from the rules:
4. End of Frame, Game or Match
(a) When Black is the only object ball remaining on the table, the first score
or foul ends the frame excepting only if the following conditions both
(i) the scores are then equal; and
(ii) aggregate scores are not relevant.
(b) When both conditions in (a) above apply:
(i) the Black is spotted;
(ii) the players draw lots for choice of playing next;
(iii) the next player plays from in-hand; and
(iv) the next score or foul ends the frame.