I heard recently from Demis Hassabis that if the hippocampus is damaged that new memories cannot be formed. I also heard recently that it has been discovered that new neurons continue to be produced in the brains of adults in the hippocampus region. Does this mean that new memories are stored in the new neurons generated in the hippocampus?
Memories are generally understood to be encoded within networks of neurons, and not within neurons themselves. New neurons are certainly useful, but they are not necessary to store new memories.
Indeed, the hippocampus is extremely plastic, meaning that the connection strength between different neurons can change rapidly, and sometimes, new connections are also formed via the growth of new synapses (this is different than the addition of new neurons). However, "new" memories are often stored using different, overlapping subsets of neurons, which is commonly referred to as global remapping in the hippocampus.
Remapping comes from the notion that many pyramidal cells in the hippocampus show place-selectivity (so-called place cells) and many of the same neurons that encode one particular environment (e.g., your bedroom), would be recruited to encode an entirely different environment (e.g, your kitchen) with no apparent relationship between the two. That is, the subset of neurons recruited to encode a particular environment seems to be random, and the topology is not preserved: neurons that encoded adjacent regions in your bedroom might encode very distant regions in your kitchen, or no regions at all.
Now to answer your question more clearly: the hippocampus is indeed critical for new memory formation; however, memories are not stored within individual neurons, but rather within collections of neurons (which could include both new and old neurons) and in the connection strengths of those neurons in particular. The same (new or old) neuron can be used and re-used in many memories, without disrupting the other memories that it forms part of.
A relatively accessible read (if you're so inclined) is Diekelmann, S., & Born, J. (2010). The memory function of sleep. Nature Reviews. Neuroscience, 11(2), 114–26. doi:10.1038/nrn2762, but it requires a paid subscription or institutional access. Alternatively, Scholarpedia has many good articles, including this one on the hippocampus, which goes a little further to explain part of our current understanding of how new memories are transferred (consolidated) to long-term storage in the neocortex.