Approximate graph pattern mining (A-GPM) is an important data analysis tool for many graph-based applications. There exist sampling-based A-GPM systems to provide automation and generalization over a wide variety of use cases. However, there are two major obstacles that prevent existing A-GPM systems being adopted in practice. First, the termination mechanism that decides when to end sampling lacks theoretical backup on confidence, and is unstable and slow in practice. Second, they suffer poor performance when dealing with the "needle-in-the-hay" cases, because a huge number of samples are required to converge, given the extremely low hit rate of their fixed sampling schemes. We build ScaleGPM, an accurate and fast A-GPM system that removes the two obstacles. First, we propose a novel on-the-fly convergence detection mechanism to achieve stable termination and provide theoretical guarantee on the confidence, with negligible overhead. Second, we propose two techniques to deal with the "needle-in-the-hay" problem, eager-verify and hybrid sampling. Our eager-verify method improves sampling hit rate by pruning unpromising candidates as early as possible. Hybrid sampling improves performance by automatically choosing the better scheme between fine-grained and coarse-grained sampling schemes. Experiments show that our online convergence detection mechanism can detect convergence and results in stable and rapid termination with theoretically guaranteed confidence. We show the effectiveness of eager-verify in improving the hit rate, and the scheme-selection mechanism in correctly choosing the better scheme for various cases. Overall, ScaleGPM achieves a geomean average of 565x (up to 610169x) speedup over the state-of-the-art A-GPM system, Arya. In particular, ScaleGPM handles billion-scale graphs in seconds, where existing systems either run out of memory or fail to complete in hours.