the number of significant figures
The significant digits you use are the values of which you are confident, plus one more value which may be uncertain.

In this case, the balance is accurate to $\pm0.001$ mg, so you're confident about the first 2 decimal places, but the 3rd place is uncertain.

The measurement given is $9.00060$, but the decimal part is smaller than our uncertainty, so we can't count on it.  The most accurate we could record is $9.001$, because the first decimal we're uncertain about is the 3rd place, which leaves us 4 significant figures.

If the measurement had been $19.00060$ instead, then the same process would result in 5 significant figures.