REST doesn't care what spelling you use for your URI.
URI design considerations typically flow from the needs of human beings working with them, rather than those of the clients (which treat the identifiers as opaque data). The structure of the URI is primarily hierarchical, with conventions that separate the hierarchical data from the non-hierarchical data. But those guidelines are human convention, much like the rules around variable names in your coding standards.
is a perfectly cromulent URI.
I was thinking to use ID 0 as "the current measurement", i.e. what the weigher is weighing at the moment of the call.
The semantics of that are perfectly reasonable. Fielding, in his thesis defining REST, wrote
The key abstraction of information in REST is a resource. Any information that can be named can be a resource: a document or image, a temporal service (e.g. "today's weather in Los Angeles"), a collection of other resources, a non-virtual object (e.g. a person), and so on. In other words, any concept that might be the target of an author's hypertext reference must fit within the definition of a resource. A resource is a conceptual mapping to a set of entities, not the entity that corresponds to the mapping at any particular point in time.
... The only thing that is required to be static for a resource is the semantics of the mapping, since the semantics is what distinguishes one resource from another.
That's not a spelling I would recommend; however. It reeks of MagicNumber. To avoid misunderstandings by human readers, I would normally go with one of the other recommendations here:
As a manner of mechanics, if the current measurement has a permanent identifier as well, I would redirect requests from the current to the permanent.
(again, in REST it is perfectly reasonable to have many resources that share the same representation, or share the same representation for the moment.)