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Ebe_Scrooge

From what you've described, I think securing it would be a sensible option.  One option would be a couple of corner braces, something like this: https://www.screwfix.com/p/corner-braces-zinc-plated-39-x-39-x-16mm-10-pack/11067

But if you really don't want to drill into the sill, then probably one of those adhesives like Gorilla Glue or Sticks Like Sh*t would be better than nothing?

[Deleted User]

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Alanp

You could try a metal frame around the window box and drill it to the walls on either side, whatever you do it needs securing, don’t use sticks like sh#t, it’s rubbish

tacpot12

If the window sill is wide enough and the window box narrow enough you could put a couple of clamps onto the front lip of the sill. Something like these would be suitable: https://www.screwfix.com/p/f-clamps-150mm-2-pack/2781v
Mount these with the handles uppermost and they will stop the box moving forward, and they can be easily removed when its time to move and they won't mark the stone. 

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ic

That looks a good size space and its flat and level.  What is immediately below the window?  Only a hurricane is going to dislodge a heavy window box.  I'd be concerned about the landlord claiming damage if you physically attached it and retaining your deposit.

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