What is surrounded by a double membrane and carries the coding that determines protein synthesis?

The structure surrounded by a double membrane that carries the coding for protein synthesis is the nucleus. The nucleus houses the cell's genetic material, DNA, which contains the instructions for synthesizing proteins. This genetic information is transcribed into messenger RNA (mRNA) before being translated into proteins by ribosomes in the cytoplasm. The double membrane, known as the nuclear envelope, protects the DNA and regulates the exchange of materials between the nucleus and the cytoplasm.