Final answer:
Convection currents will stop when the ice cube completely melts and the water reaches thermal equilibrium, as there will be no temperature gradient to maintain the convection.
Step-by-step explanation:
Convection currents in the beaker will stop when the ice cube completely melts and the temperature throughout the beaker equalizes, resulting in no temperature gradient to drive the convection. This occurs because as the ice melts, it absorbs heat energy without increasing in temperature due to the heat of fusion. Once all the ice has melted, the system can eventually reach thermal equilibrium where the liquid's temperature evens out, and without a temperature difference, convection currents cease.