Not only do Africa's fragile states grow more slowly than non-fragile states, but they seem to be caught in a "fragility trap". For instance, the probability that a fragile state in 2001 was still fragile in 2009 was 0.95. This paper presents an economic model where three features -- political instability and violence, insecure property rights and unenforceable contracts, and corruption -- conspire to create a slow-growth-poor-governance equilibrium trap into which these fragile states can fall. The analysis shows that, by addressing the three problems, fragile countries can emerge from the fragility trap and enjoy a level of sustained economic growth. But addressing these issues requires resources, which are scarce because external aid is often tailored to the country's performance and cut back when there is instability, insecurity, and corruption. The implication is that, even if aid is seemingly unproductive in these weak-governance environments, it could be hugely beneficial if it is invested in such a way that it helps these countries tackle the root causes of instability, insecurity, and corruption. Empirical estimations corroborate the postulated relationships of the model, supporting the notion that it is possible for African fragile countries to avoid the fragility trap.