Late nineteen-teens to about 1920.
Although this apron is seamless, you're still going to have to piece the fabric if you're using narrow fabric, but otherwise, this is a very simple pattern, and probably very popular.
Although there is a certain satisfaction in purchasing a pristine, unused pattern, patterns that have seen some use have stories to tell us. This pattern is well-used, with multiple tears in the pattern tissue and one small torn-off piece that was carefully pinned back on. If the pattern pieces could speak, I'd love to know how many times this apron pattern was made up.
The maker has penciled in a shorter cutting line, as well as the line along the side where the apron will need to be pieced.
It's a little hard to see but you can just see that the artist has indicated rick-rack trim on the short version:
Rick-rack seems to have been a very common trimming for aprons. Nu-fashond rick-rack was a common brand in the 1920s. Sewing patterns from this period seldom mention notions and trimming, but sewing books of the period frequently mention trimming house or bungalow aprons with rick-rack.
6/18/2010 - Update. I had a free Saturday so I took the time to make this up in a very cheery remnant I found last year. Fortunately, because the print is so incredibly busy, you don't really notice that I didn't have enough material to match the side pieces or the pockets (if you look closely, though, you can see where the trelliswork doesn't exactly match.) Unfortunately, I see now that the print wasn't precisely centered on the fabric, so the design is about 3 inches off center, darn it. This fabric was 45 inches wide, so I had to piece the sides. I finished these seams with a flat fell.
The approximate circumference at the bottom of the armholes is 36 inches.
The very brief instructions indicate that if preferred, you may underface the edges, so I think the expectation would be that you would bind them. I decided to underface with lavender gingham bias that I'd cut for the purpose. Here you can see that I've pinned the folded bias on, but basted it through the deep front curve, since I think that gives me more control as I'm sewing.