The mass of one mole of a substance is the mass of Avogadro's number ($N_A$) of its constituent particles (atoms, molecules, etc.) . $\text{Mass of 1 mole of carbon} = \text{Mass of 1 carbon atom} \times N_A$. If the value of Avogadro's number ($N_A$) is changed from $6.022 \times 10^{23} \text{ mol}^{-1}$ to $6.022 \times 10^{20} \text{ mol}^{-1}$, the number of carbon atoms present in one mole will decrease. Consequently, the mass of one mole of carbon will also change (it will decrease by a factor of $10^3$).

The definition of mass in units of grams, the ratio of chemical species to each other in a balanced equation, and the ratio of elements to each other in a compound are fundamental concepts and do not depend on the specific value of Avogadro's number.