The usefulness of graphene for electronics has been limited because it does not have an energy bandgap. Although graphene nanoribbons have non-zero bandgaps, lithographic fabrication methods introduce defects that decouple the bandgap from electronic properties, compromising performance. Here we report direct measurements of a large intrinsic energy bandgap of approximately 50 meV in nanoribbons (width, approximately 100 nm) fabricated by high-temperature hydrogen-annealing of unzipped carbon nanotubes. The thermal energy required to promote a charge to the conduction band (the activation energy) is measured to be seven times greater than in lithographically defined nanoribbons, and is close to the width of the voltage range over which differential conductance is zero (the transport gap). This similarity suggests that the activation energy is in fact the intrinsic energy bandgap. High-resolution transmission electron and Raman microscopy, in combination with an absence of hopping conductance and stochastic charging effects, suggest a low defect density.