Canon:

Arthur was one of three brothers and had no sister. We don't know his brothers' names.
Molly's maiden name was Prewett and she had at least 2 brothers, Gideon and Fabian. We don't know if Molly had sisters; none are ever mentioned.
Ron (and by extension Bill, Charlie, Percy, Fred, George and Ginny) had an uncle Bilius, a great-aunt Muriel and a grandfather who played Wizard Chess. In addition to uncles Gideon and Fabian, of course.
Ginny is the only Weasley girl for generations.
Molly refers to a relative as "Our Great Auntie Muriel," but because she says "our," this could mean Muriel was a great aunt of Molly and Arthur, or of Ron and his siblings.
Molly and Sirius are "cousins by marriage."
Therefore, Molly's parents are probably closely related to the 'Ignatius Prewett' listed on the Black Family tree. If the Black Family tree Jo auctioned is complete, Ignatius and Lucretia cannot be Molly's parents themselves because they are indicated as being childless.
Arthur is "something like" Sirius's second cousin once removed. This describes a relationship where one person's great-great grandparent is the same person as the other person's great grand-parent.
Therefore, Septimus Weasley is Arthur's father. Septimus's children would indeed Sirius's second cousins once removed. Assuming the Black family tree Jo auctioned is complete, this relationship fits the facts and can be explained no other way.

Guesswork (not canon):

Bilius was one of Arthur's 2 brothers. Not canon: We don't know Bilius's surname and we don't know if Molly had more than 2 brothers. "Uncle" can also be an honorary name given to non-family members, though this doesn't seem likely.
The grandfather who gave Ron the chess set is Septimus Weasley, Arthur's father. Not canon: We don't know his surname, so it could have been either Arthur's father Septimus, or Molly's father.