The intriguing properties of graphene have inspired the pursuit of two-dimensional materials with honeycomb structure. Here we achieved the synthesis of a monolayer transition-metal monochalcogenide AgTe on Ag(111) by tellurization of the substrate. High-resolution scanning tunneling microscopy, combined with low-energy electron diffraction, angle-resolved photoemission spectroscopy, and density functional theory calculations, demonstrates the planar honeycomb structure of AgTe. The first-principles calculations further predict that, protected by the in-plane mirror reflection symmetry, there are two Dirac node-line fermions existing in the free-standing AgTe when spin-orbit coupling (SOC) is ignored. In fact, the SOC leads to the gap opening, resulting in the emergence of the topologically nontrivial quantum spin Hall edge state. Importantly, our experiments evidence the chemical stability of the monolayer AgTe in ambient conditions, making it possible to study AgTe by more ex situ measurements and even to utilize AgTe in future electronic devices.