Given:

Dipole moment, $ |\vec{P}| = 5 \times 10^{-6} \, \text{Cm} $

Electric field magnitude, $ |\vec{E}| = 4 \times 10^5 \, \text{N/C} $

Initial angle, $ \theta_i = 0^\circ $

Final angle, $ \theta_f = 60^\circ $

To calculate the change in potential energy $ \Delta U $ of the dipole:

$ \Delta U = U_f - U_i = -PE \cos \theta_f + PE \cos \theta_i $

Simplifying, we have:

$ \Delta U = PE \left( \cos \theta_i - \cos \theta_f \right) $

Substitute the known values:

$ \Delta U = 5 \times 10^{-6} \times 4 \times 10^5 \left(1 - \frac{1}{2}\right) $

Calculate further:

$ \Delta U = 5 \times 10^{-6} \times 4 \times 10^5 \times \frac{1}{2} $

$ \Delta U = 10 \times 10^{-6} \times 10^5 $

Which simplifies to:

$ \Delta U = 1 \, \text{J} $

Thus, the change in the potential energy of the dipole is $\boxed{1 \, \text{J}}$.