  We prove the following results: (1) There is a one-relator inverse monoid
$\mathrm{Inv}\langle A\:|\:w=1 \rangle$ with undecidable word problem; and (2)
There are one-relator groups with undecidable submonoid membership problem. The
first of these results answers a problem originally posed by Margolis, Meakin
and Stephen in 1987.
