This is my favorite baby bib to crochet. You can find the pattern HERE. You can't tell from my pic but it has a fpdc design in it. Its more appearant in the photo on the pattern page. But I like it because it doesn't have ties. Its easy to make....only takes a couple hours. Its not a small bib....it would be too big for a newborn, but an older baby starting on solids would benefit from it.
My b/f visited his little grandson William the other day and the baby's father said they didnt have many bibs for him. So I made this one for him today. Tomorrow my b/f will deliver the bib. I bought some extra cotton ww yarn today to make more.
I havent tried, but you could probably use a finer weight cotton and a smaller hook to make one for a newborn. I might just have to try it. :)