Thanks for the replies. To clarify — is this not a measurement of the two particles' masses to ~38 significant figures?

None of those zeros are significant figures.

I wondered from the announcement whether the other figures would be present somewhere in the paper.

Measured in grams like the 10^-38, there would still be lots of zeros first, so not 38 significant figures.

Besides, relative measurements are often easier than absolute ones, so even if you could determine that the masses of two objects of ~1g differed by 10^-38g, that wouldn't necessarily tell you the absolute masses to anywhere near that accuracy.