In a triangle inscribed in a circle where one side (in this case, line \( AB \)) is the diameter, the angle opposite to that side (angle \( C \)) is a right angle (90°). This is known as the Thales' theorem.
Given that angle \( A \) measures 15°, we can find angle \( B \) using the fact that the angles in a triangle sum to 180°:
\[ \angle A + \angle B + \angle C = 180° \]
Substituting the known values:
\[ 15° + \angle B + 90° = 180° \]
This simplifies to:
\[ \angle B = 180° - 15° - 90° = 75° \]
Now, to find the arc measure of \( AC \), we note that the measure of the arc subtended by angle \( A \) at the center of the circle is twice the measure of angle \( A \), due to the property of inscribed angles. Since \( \angle A \) subtends the arc \( AC \),
\[ \text{Arc } AC = 2 \cdot \angle A = 2 \cdot 15° = 30°. \]
Thus, the arc measure of \( AC \) is \( \boxed{30°} \).