  We study the transport properties of a quantum dot (QD) with highly resistive
gate electrodes, and show that the QD displays a quantum phase transition
analogous to the famous dissipative phase transition first identified by S.
Chakravarty [Phys. Rev. Lett. {\bf 49}, 681 (1982)]; for a review see [A. J.
Leggett {\em et al.}, Rev. Mod. Phys. {\bf 59}, 1 (1987)]. At temperature T=0,
the charge on the central island of a conventional QD changes smoothly as a
function of gate voltage, due to quantum fluctuations. However, for
sufficiently large gate resistance charge fluctuations on the island can freeze
out even at the degeneracy point, causing the charge on the island to change in
sharp steps as a function of gate voltage. For $R_g<R_C$ the steps remain
smeared out by quantum fluctuations. The Coulomb blockade peaks in conductance
display anomalous scaling at intermediate temperatures, and at very low
temperatures a sharp step develops in the QD conductance.
