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Open the casement window. Measure across the width of the window opening from the inside track where the sash meets the frame when it's closed. Subtract 1/2 inch from the measurement. Do the same for the height of the opening. Order the replacement window based on these measurements.
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Pry off the thin trim that borders the interior edge of the opening, using your hammer and pry bar. Keep the trim intact as you take it out. Remove the crank mechanism and other hardware of the casement window from the surrounding frame, using a screw gun. Pull out the window.
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Cut around the line where you removed the trim, where the casing meets the frame, using a reciprocating saw with a metal-cutting blade. Cut through the screws that hold the casing into the frame. Pull the frame out.
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Set the double-hung window unit into the opening. Set the bottom of the window onto the sill. Push the rest of window up and in, until it rests against the trim that borders the exterior edge of the opening.
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Hold your level alongside the unit to check for level. Slide shims between the window and the frame until the unit is plumb -- standing perfectly vertical. Check around the window with the level to make sure it's plumb and level, shimming as needed.
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Secure the window in place by sinking long mounting screws, usually provided with the window unit, though the holes at the sides of the unit and into the surrounding frame, using the screw gun.
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Spray low-expansion foam insulation around the edges of the unit, into the space between the casing and the surrounding wall. Reinstall the interior border of trim to hide the gap, using a hammer and finishing nails. Apply a bead of caulk around the trim and the window.
Check out this related video from Homesteady on Youtube.