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DNA-stabilized silver nanoclusters (AgN-DNAs) are a class of nanomaterials comprised of 10-30 silver atoms held together by short synthetic DNA template strands. AgN-DNAs are promising biosensors and fluorophores due to their small sizes, natural compatibility with DNA, and bright fluorescence---the property of absorbing light and re-emitting light of a different color. The sequence of the DNA template acts as a "genome" for AgN-DNAs, tuning the size of the encapsulated silver nanocluster, and thus its fluorescence color. However, current understanding of the AgN-DNA genome is still limited. Only a minority of DNA sequences produce highly fluorescent AgN-DNAs, and the bulky DNA strands and complex DNA-silver interactions make it challenging to use first principles chemical calculations to understand and design AgN-DNAs. Thus, a major challenge for researchers studying these nanomaterials is to develop methods to employ observational data about studied AgN-DNAs to design new nanoclusters for targeted applications. In this work, we present an approach to design AgN-DNAs by employing variational autoencoders (VAEs) as generative models. Specifically, we employ an LSTM-based β-VAE architecture and regularize its latent space to correlate with AgN-DNA properties such as color and brightness. The regularization is adaptive to skewed sample distributions of available observational data along our design axes of properties. We employ our model for design of AgN-DNAs in the near-infrared (NIR) band, where relatively few AgN-DNAs have been observed to date. Wet lab experiments validate that when employed for designing new AgN-DNAs, our model significantly shifts the distribution of AgN-DNA colors towards the NIR while simultaneously achieving bright fluorescence. This work shows that VAE-based generative models are well-suited for the design of AgN-DNAs with multiple targeted properties, with significant potential to advance the promising applications of these nanomaterials for bioimaging, biosensing, and other critical technologies.