JSON.stringify throws RangeError: Invalid string length for huge objects

As the title implies I'm trying to stringify huge JavaScript Object with JSON.stringify in my Node.js app. The objects are - again - huge (tens of mega bytes), they don't contain any functions. I need to write the serialized objects to a file. What I'm getting now is this:

RangeError: Invalid string length
  at Object.stringify (native)
  at stringifyResult (/my/file.js:123:45) -> line where I use JSON.stringify

Any idea how to solve that issue?