The two most prominent and ubiquitous features of glasses at low temperatures, namely the presence of tunneling two-level systems and the so-called boson peak in the reduced vibrational density of states, are shown to persist essentially unchanged in highly stabilized glasses, contrary to what was usually envisaged. Specifically, we have measured the specific heat of 110 million-year-old amber samples from El Soplao (Spain), both at very low temperatures and around the glass transition T-g. In particular, the amount of two-level systems, assessed at the lowest temperatures, was surprisingly found to be exactly the same for the pristine hyperaged amber as for the, subsequently, partially and fully rejuvenated samples.