The Dark Silicon provides opportunities to realize Reliability- Heterogeneous Processors with ISA compatible cores having different levels of protection against reliability threats (like soft errors). This paper presents design-time customization of Reliability-Heterogeneous Processors given a set of applications and area constraints. A run-time system adaptively manages the soft error resilience under a given thermal design power (TDP) budget. We synthesize an embedded processor with different levels of protection and present area and power results for a 45nm technology. We illustrate the benefits of adaptive soft error resilience by comparing it with four different state-of-the-Art approaches where we achieve 58%-96% overall system reliability improvements under a tight TDP constraint (corresponding to a 65% dark area).