WARNING. Ice cream is OK (because of oils in it I suspect), but popsicles can be DANGEROUS. I speak from experience.
I used to always buy a popsicle on my cycle commute. In the winter, I could put it in my pocket, and it wouldn't melt. It was run to ride across the parking lot in the snow, eating a popsicle as I arrived for work.
It isn't fun at -20F.
Know how your mother told you never to lick a pole in the cold? Same thing happens with a popsicle at -20F.
One cold day, I put a popsicle in my mouth, and my tongue, upper and lower lip froze into one solid piece. IT IS PAINFUL! I ran screaming (MFMFMFMF!) into the nearest bathroom, and stuck my head under the sink to defrost it. (Getting strange looks in the process) Lips swollen for a few days.
BE careful what you eat at very cold temperatures!