your comment "only a single record of the first table can be referenced via a foreign key in the second" is true for one-to-many tables as well
in a one-to-one relationship, at most one record of the second table can have a foreign key referencing any single record in the first table
for mysql, use innodb tables in order to get foreign key support
to restrict the relationship to one-to-one, you would use a CHECK constraint, and i don't think mysql supports it