In the vacuum of space the most important consideration is to consider how much radiation an ice cube would absorb from, for example, nearby stars and how fast the ice cube itself would radiate away energy (using Wien's law), finding what ice cube temperature would produce an equilibrium (the temperature at which the ice cube radiate energy at the same rate it absorbed energy) and then determining if that temperature is above or below the melting point of the ice cube. If it is above the melting point (of water in a vacuum), then as the other answers have said the ice cube would sublimate; if it is below the melting point then the ice cube would stay frozen.
Specifically for an ice cube that is a cube in orbit around the sun with one side facing the sun you would need to calculate how much energy the side facing the sun absorbs from the sun as well as how much energy radiates away from all six sides of the cube and then find the equilibrium temperature.