To solve the problem, we start by finding the measure of the interior angles of a regular hexagon (which is a polygon with six sides and equal angles).
The formula for the interior angle of a regular polygon is given by:
\[ \text{Interior angle} = \frac{(n - 2) \times 180^\circ}{n} \]
where \( n \) is the number of sides of the polygon. For a hexagon, \( n = 6 \):
\[ \text{Interior angle} = \frac{(6 - 2) \times 180^\circ}{6} = \frac{4 \times 180^\circ}{6} = \frac{720^\circ}{6} = 120^\circ \]
Now, we will calculate the corresponding exterior angle at one of the vertices of the hexagon. The exterior angle can be found using the formula:
\[ \text{Exterior angle} = 180^\circ - \text{Interior angle} \]
Substituting the interior angle we found:
\[ \text{Exterior angle} = 180^\circ - 120^\circ = 60^\circ \]
Thus, the measure of the exterior angle at that vertex is:
\[ \boxed{60 \text{ degrees}} \]