Please help me identify this vintage Rolex?

[tspero]

I own a vintage Rolex which was inherited from my grandfather, I believe from the 1930s or 1940s. I have looked through a few Rolex books and online auctions, but have been unable to determine what the model is and what it's production date was. Due to it's age I am hesitant to open the case. It still keeps perfect time which is amazing for the age! I believe the gold band is aftermarket since it does not have a Rolex marking. The leather band and case both appear to be original.

Anyone who can help me identify this watch would be greatly appreciated, and before anyone asks, I am not interested in selling it.

Thanks in advance!