The Academies Enterprise Trust (AET), which runs Weston Academy, said it made the "sad decision", due to falling pupil numbers.
The 56-pupil school was classed as "inadequate" by Ofsted in March.
A petition set up by parent George Metcalf, describing closing the school as a "huge waste of resources" attracted more than 500 signatures.
The campaign won the backing of the National Secular Society as parents claimed the nearest alternatives were all faith schools.
The school was placed in special measures following an Ofsted report in March which criticised leadership and management, pupils' behaviour, the quality of teaching and achievement of pupils.
But in May the watchdog reported that teaching and leadership was improving.
AET said the school had less than 40 pupils and the Department for Education had confirmed its agreement to closing the school at the end of December.
Chief executive Ian Comfort said: "Closing a small school is always a very sad decision to make, and is only taken as a very last resort.
"The declining numbers at Weston, however, have been making it increasingly difficult for the school to provide pupils with the quality of education they deserve. I am sure they will all thrive in their new schools and we wish them, and the staff, every success in the future."
An Isle of Wight primary school is to close, despite a parents' campaign to keep it open.