what's the meaning of "frozen over"?

[keannu]

What is the meaning of "over" in this sentence?

It must be very cold outside, for the lake is frozen over.

Does it mean all over the place? or just above?

 

[jlinger]

The surface is frozen, from one shore to the other, but not the entire depth of the water in the lake. That is, you could walk across the lake, but fish can still swim in it.