Converting a 2d surface into a 3d volume

I have a 2d surface shown in the image below using ListPlot3D[]:

image It is generated with one line of code:

ListPlot3D[RandomVariate[UniformDistribution[], {10,10}]]     

Currently, it’s a ‘white noise’ surface, meaning the surface is more or less random between zero and one. I’d like to give this thickness (so I can ultimately 3d print it).

I’d like the remaining 5 sides to be flat, so that I’d end up with a "cube" with one surface jagged.

Is there an easy way to do this? Essentially just fill in everything below the surface?