“I recently lost my engagement ring. (It's not the first time I lost it, but I think this time it's really gone. Sigh.). But, I’m not alone—as I just learned that Jupiter lost a band recently… How many times bigger is Jupiter's band than my engagement band?”
Jupiter’s radius is about 71,000 km.1 My sister’s ring’s radius was likely about 1.0 cm. This would make Jupiter’s ring (or band) about 7 billion times bigger if we’re talking about its linear dimension. However, the word “bigger” has several meanings. If we’re talking about how many times larger the volume of the ring is, then we need to cube the linear dimension. In this case, Jupiter’s ring is about 3.6×1029 times bigger. As antoniseb has pointed out, there’s also the issue of density. If we want to talk about how much more massive the ring is we need to know the ratio of Jupiter’s ring’s density to the density of gold. The density of gold is 19.3 g/cm3. Jupiter is a gas giant with a low average density of about 1.33 g/cm3. The upper atmosphere where the ring is located is probably much less dense. Assuming it has the same density as air at the surface Earth (~0.0012 g/cm3), Jupiter’s band will be about 2.2×1025 times bigger.
Sorry about the ring. If you want to make sure you don’t lose your next ring, you can always do what I did.2
 Since the ring is not at the equator, it will actually be a little smaller than this. See picture above.