In a triangle, the sum of the measures of the angles is always \(180^\circ\). Given that \(m\angle A = 63^\circ\) and \(m\angle C = 99^\circ\), we can find the measure of angle \(B\) using the following formula:
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting the given values:
\[ 63^\circ + m\angle B + 99^\circ = 180^\circ \]
Now, combine the angles \(A\) and \(C\):
\[ 162^\circ + m\angle B = 180^\circ \]
Next, isolate \(m\angle B\):
\[ m\angle B = 180^\circ - 162^\circ \]
Calculating this gives:
\[ m\angle B = 18^\circ \]
Thus, the measure of angle \(B\) is \( \boxed{18^\circ} \).