The organelle in the eukaryotic cell nucleus that produces ribosomes is called the nucleolus.

Here's why:

* Ribosomes are essential for protein synthesis: They are the sites where mRNA is translated into protein.

* Ribosomes are made of rRNA and proteins: The rRNA is transcribed in the nucleolus, and the proteins are transported from the cytoplasm.

* The nucleolus is a dense region within the nucleus: It is not membrane-bound, but it is a distinct structure where ribosome assembly takes place.

So, the nucleolus acts like a factory for ribosomes, ensuring that the cell has a constant supply of these essential protein-making machines.