On newer digital meters, a variety of techniques are used. The most common one is called dual-slope integration. There the incoming voltage is again fed into a resistor to convert the voltage to a current. Then this current is used to discharge a capacitor until the voltage gets down to zero. Then a known current is applied, and the time it takes to recharge the capacitor is a measure of the input curent.

A faster way is to apply the input voltage to a A/D converter which uses a successive-approximation technique, a version of the old "20-questions" game. First it compares the input voltage to 1/2 the maximum and decides whether the input is above or below that, then it splits the difference, choosing either 1/4 or 3/4, comparing that, then halving the interval again. After 16 such comparisons, you have the answer to within 4+ decimal digits (1 part in 65,536), after 24 comparisions you have 7+ digits(1 part in 16,000,000). That's usually more than plenty of accuracy.