It's a simple case of the small cog being worn too much to mesh with the new chain properly.

As [MENTION=20650]T-Mar[/MENTION] pointed out, many bikes have spent their life primarily in the smallest freewheel cog, resulting in premature wear of that particular cog. This skipping is a common result when a new chain is installed without changing the freewheel (or spinning on a new small cog onto the existing freewheel, if one is lucky enough to have a freewheel that can be dismantled + a new replacement cog). Also, since these cogs have the smallest amount of chain wrap, they're more susceptible to wear than their larger counterparts. Plus, the chain is under the least amount of tension in the small/small combo.

This isn't unusual, especially when throwing a new chain on an old freewheel. To make it even more difficult, sometimes it is next to impossible to see any wear on a worn small cog. It seems to baffle quite a few mechanics.

Also, to address a few alternate diagnosis here and why they're not the problem:

• It's not a stiff link. The chain wouldn't rise off the freewheel the same way, and you'd see the derailer reacting to the stuck link as it threads through the pulley cage. It would also require a full revolution for a second pop.
• It's not chainline. By the time the chain gets fed through the upper jockey wheel, said jockey wheel is negating the poor chainline. Also, the chain would be trying to pop off the cog at around 12-o'clock on the top if so. It's currently trying to pop at 9-o'clock.
• The chain is not trying to shift into the larger cog. There's absolutely no side-to-side play visible when it pops. Plus, that's a 5-speed, non-ramped freewheel being run with what appears to be a modern 7-speed chain. There are oodles of space between cogs. Any attempt at ghost shifting would be painfully obvious.

Replace the freewheel and you'll be fine.

-Kurt