Many parents are set to be "disappointed" today when their children are rejected for the school of their choice, a Government minister has warned.
Priti Patel said it was "deeply regrettable" that families in England would be hit by a shortage in primary school places.
The employment minister, a Brexit supporter, blamed uncontrolled migration for putting "unsustainable pressure" on public services and warned the problem would only get worse as more countries became members of the European Union.
Hundreds of thousands of four and five-year-olds will learn which primary school they will be attending from September, on what is commonly known as National Offer Day.
Ms Patel said: "The shortage of primary school places is yet another example of how uncontrolled migration is putting unsustainable pressures on our public services.
"Education is one of the most important things that Government delivers, and it's deeply regrettable that so many families with young children are set to be disappointed today.
"The truth is that for as long as we remain a member of the EU we are completely unable to control the numbers of people coming to this country - and with another five countries in the pipeline to join the EU, the problem is set to get even worse.
"If we Vote Leave we can take back control of our borders."
Unions and town hall leaders warned Government reforms that mean all schools will convert to academies are set to fuel the shortage in school places.
Councils will not have the power to force schools to expand in the future, even where there is demand and capacity, the Local Government Association (LGA) said.
It warned that an additional 336,000 primary school places would be needed by 2024.
The Association of Teachers and Lecturers (ATL) said it was "worried" about the impact of the reforms.
But a Whitehall source said: "There's no evidence that migration is the key driver of demand for primary school places."