Bubble bursting at liquid surfaces is ubiquitous and plays a key role for the mass transfer across interfaces, impacting global climate and human health. Here, we document an unexpected phenomenon that when a bubble bursts at a viscoelastic surface of a bovine serum albumin solution, a secondary (daughter) bubble is entrapped with no subsequent jet drop ejection, contrary to the counterpart experimentally observed at a Newtonian surface. We show that the strong surface dilatational elastic stress from the viscoelastic surface retards the cavity collapse and efficiently damps out the precursor waves, thus facilitating the dominant wave focusing above the cavity nadir. The onset of daughter bubble entrainment is well predicted by an interfacial elastocapillary number comparing the effects of surface dilatational elasticity and surface tension. Our Letter highlights the important role of surface rheology on free surface flows and may find important implications in bubble dynamics with a contaminated interface exhibiting complex surface rheology.