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Xanathar's Guide to Everything adds an option to have a weapon made of or coated with adamantine for an additional 500 GP above the weapon's base price (XGtE, 78). How does this interact with the Player's Handbook option of having a weapon coated with silver for an additional 100 GP above the weapon's base price (PHB, 148)?
Can a weapon be both adamantine and silvered?
Yes. A weapon can be made of adamantine, but coated with silver.
The description of silvered weapons notes that the price includes not only attaching silver to the weapon, but doing so in a way that does not reduce its effectiveness. Therefore, if you silver an adamantine weapon (nothing in the description of silver limits it to steel weapons, so technically you can even silver a wooden weapon), it still retains the benefit of being adamantine.
Nothing in the rules as they currently exist bars silvering an adamantine weapon, or any weapon of any material.
However, given that the XGtE was published after the PHB, there is a possibility that the game designers did not really think carefully about this combination. The silvering process is not clearly detailed, meaning that the way it is performed is left to the gamers. The actual answer might depend on the metaphysics of your own gaming world (meaning your DM).
If your fantasy world has similar chemistry/physics as our daily world (which is not an unreasonable assumption for ordinary objects: swords are still metal for example, or gold is malleable so you can make coins, etc), you need to make a decision about what sort of material adamantine is. If the "silvering process" on metals is similar to silver plating, there are some good news: silver can be deposited on nearly any metal. Moreover, while silver plating technology was developed in the 18th century in the real world, so-called Sheffield plating is simple enough that you could imagine it was invented earlier in your fantasy world. If you decide that adamantine does act as a good substrate (like copper), Sheffield plating becomes possible. On the other hand, if you decide it is not a good substrate, you need the invention of electroplating (or the fantasy equivalent of that) to silverplate adamantine.
