This paper shows that the elimination of fault-agnostic instability, the instability caused by fault-agnostic distributed control, substantially improves BGP convergence speed. To this end, we first classify BGP convergence instability into two categories: fault-agnostic instability and distribution-inherent instability; secondly, we prove the impossibility of eliminating all distribution-inherent instability in distributed routing protocols; thirdly, we design the Grapevine Border Gateway Protocol (G-BGP) to show that all fault-agnostic instability can be eliminated. G-BGP eliminates all fault-agnostic instability under different fault and routing policy scenarios by (i) piggybacking onto BGP UPDATE messages fine-grained information about faults to the nodes affected by the faults, (ii) quickly resolving the uncertainty between link and node failure as well as the uncertainty of whether a node has changed route, and (iii) rejecting obsolete fault information. We have evaluated G-BGP by both analysis and simulation. Analytically, we prove that, by eliminating fault-agnostic instability, G-BGP achieves optimal convergence speed in several scenarios where BGP convergence is severely delayed (e.g., when a node or a link fail-stops), and when the shortest-path-first policy is used, G-BGP asymptotically improves BGP convergence speed except in scenarios where BGP convergence speed is already optimal (e.g., when a node or a link joins). By simulating networks with up to 115 autonomous systems, we observe that G-BGP improves BGP convergence stability and speed by an order of magnitude.