Yes, funeral insurance normally covers all costs for a
funeral.
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The estate pays for the funeral. It is typically one of the
first clauses in the will. If someone signed for the funeral costs,
they can be held liable if the estate fails to cover the costs.
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If there is no money to pay funeral costs in the estate the
executor is not personally responsible. The estate can be sued to
have any assets sold and the funeral home is paid first.
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The estate has the responsibility to pay off the funeral costs
and debts. If the estate cannot do so, they distribute as best they
can. If the court approves the distribution, the debts are
ended.