I've been thinking of having an underwater structure in a world with ice, but I would think water from currents would melt the ice. Would there be any way to keep it frozen and have an underwater structure of some sort?
Salt water has a lower freezing temperature than distilled water. The freezing point for fully saturated salt water is about -21 C. Other impurities could lower it even further. If the ocean temperature was at -10 C, the salt water would not freeze, but already frozen fresh water ice would not melt.
The bigger problem is how to keep it submerged, because ice floats.
very cold (fresh) water and ice nucleating agents (proteins, etc.) maybe. mix in some endothermic reactions on the inside of the underwater castle to keep things cold enough to maintain the ice (As guideline if you're looking for some reactions, water needs to lose 333.55 kJ of energy per kg of water to freeze 0 degree water into 0 degree ice.)
edit: also is the structure filled with water? because if not, you have to make it strong enough to withstand the pressure of the lake water pushing in against flimsy ol' air.
Methane clathrate (CH4·5.75H2O) or (4CH4·23H2O), also called methane hydrate, hydromethane, methane ice, fire ice, natural gas hydrate, or gas hydrate, is a solid clathrate compound (more specifically, a clathrate hydrate) in which a large amount of methane is trapped within a crystal structure of water, forming a solid similar to ice. Originally thought to occur only in the outer regions of the Solar System, where temperatures are low and water ice is common, significant deposits of methane clathrate have been found under sediments on the ocean floors of the Earth. Methane clathrates are common constituents of the shallow marine geosphere and they occur in deep sedimentary structures and form outcrops on the ocean floor. Methane hydrates are believed to form by the precipitation or crystallisation of methane migrating from deep along geological faults. Precipitation occurs when the methane comes in contact with water within the sea bed subject to temperature and pressure.
Methane clathrates have methane molecules incorporated into water ice. Other hydrocarbon gas/water ice clathrates are possible too. They require cold and pressure. Depicted are the arms of a remote control submarine messing with a piece they found. This ice can accumulate in big outcrops. Interestingly the ice is always covered with life; there is some sort of metabolism going on that I have never seen laid out, probably with methanotrophs at the base who are happy to have their dinner held still for them by the clathrate.