Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two

We classify all noetherian Hopf algebras $H$ over an algebraically closed field $k$ of characteristic zero which are integral domains of Gelfand-Kirillov dimension two and satisfy the condition ${\text{Ext}}_H^1(k,k)\mathrm{\ne }0$. The latter condition is conjecturally redundant, as no examples are known (among noetherian Hopf algebra domains of GK-dimension two) where it fails.