To solve the problem, we begin with the information given.
- The measurement of angle ABC is 68 degrees.
- Maria constructs the angle bisector BD of angle ABC.
When BD bisects angle ABC, it divides it into two equal angles. Therefore:
\[ \text{Measurement of angle ABD} = \text{Measurement of angle DBC} = \frac{68^{\circ}}{2} = 34^{\circ} \]
Next, Maria constructs the angle bisector BE of angle DBC.
Since we found that angle DBC measures 34 degrees, when BE bisects angle DBC, it also divides it into two equal angles:
\[ \text{Measurement of angle EBC} = \frac{34^{\circ}}{2} = 17^{\circ} \]
Thus, the measurement of angle EBC is:
\[ \boxed{17 \text{ degrees}} \]