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System.DllNotFoundException on MonoDevelop
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I have not had that specific problem, but I'm going to venture a guess as to its cause and suggest a remedy based on that guess. If the remedy solves your problem, the guess is confirmed; otherwise that experiment may inform later investigation.

My guess is that System.Data.SQLite.dll is not located in a place known to the running Mono VM when it runs your executable, and therefore cannot be loaded. The remedy is to locate that DLL and place it or a copy in the same directory as your application executable.