Thermal conductors don't have to be hot to transfer heat, explain a situation when a ice cube would still transfer heat to another object it is in contact with.

An ice cube would transfer heat to another object whose temperature
is lower than zero°C (32°F).

A block of "dry ice" is sitting there at a temperature of -78°C (-109°F).
An ice cube helps to melt dry ice nice and fast.

If you could find a block of solid nitrogen, its temperature would be
63K (-210°C, -346°F).  An ice cube would transfer heat to that baby
so fast that it would instantly boil.