‘[Egyptian housewife Ghazala Ibrahim] is not blessed with one but nine babies’ (New Paper, 4 August 2008).
The pair not ... but should mark out the sentence parts that are being contrasted. In the New Paper’s caption, the misplaced not seemingly contrasts blessed with something else: cursed, for instance (i.e. she is not blessed, but cursed).
Hence, make it: ... is blessed with not one but nine babies. Note the contrast between one and nine. (Alternatively, we can say ... is blessed not with one but (with) nine babies — the second with may be ellipted, or left out because it is redundant.)