  We study the conductivity of graphene with a smooth but
particle-hole-asymmetric disorder potential. Using perturbation theory for the
weak-disorder regime and numerical calculations we investigate how the
particle-hole asymmetry shifts the position of the minimal conductivity away
from the Dirac point $\varepsilon = 0$. We find that the conductivity minimum
is shifted in opposite directions for weak and strong disorder. For large
disorder strengths the conductivity minimum appears close to the doping level
for which electron and hole doped regions ("puddles") are equal in size.
