No present observations suggest a technologically advanced extraterrestrial intelligence (ETI) has spread through the galaxy. However, under commonplace assumptions about galactic civilization formation and expansion, this absence of observation is highly unlikely. This improbability is the heart of the Fermi Paradox. The Fermi Paradox leads some to conclude that humans have the only advanced civilization in this galaxy, either because civilization formation is very rare or because intelligent civilizations inevitably destroy themselves. In this paper, we argue that this conclusion is premature by introducing the "Sustainability Solution" to the Fermi Paradox, which questions the Paradox's assumption of faster (e.g. exponential) civilization growth. Drawing on insights from the sustainability of human civilization on Earth, we propose that faster-growth may not be sustainable on the galactic scale. If this is the case, then there may exist ETI that have not expanded throughout the galaxy or have done so but collapsed. These possibilities have implications for both searches for ETI and for human civilization management.