The number of EU citizens moving to the UK has continued to drop, but more people coming from elsewhere means the overall migration rate is unchanged.
Figures show net migration - the difference between how many people came to the UK for at least 12 months and how many left - was 273,000 last year.
EU net migration was 74,000 in the year to the end of June 2018, while non-EU net migration was 248,000.
The ONS said more Asian citizens had been moving to the UK for work.
The Office for National Statistics says 219,000 EU citizens arrived in the UK, as 145,000 left, making net EU migration the lowest it had been since 2012.
The data also showed non-EU net migration was the highest since 2004.
Jay Lindop, director of the ONS's centre for international migration, said overall net migration had peaked in 2016 and remained fairly stable since then.
Overall net migration takes into account the migration of British citizens, of which 49,000 more left the country than returned in the year to the end of June 2018.
In the year to the end of June 2017, EU net migration was 107,000 and non-EU net migration was 173,000. Overall net migration was 273,000 in the same period.
The government's policy is to reduce overall migration to below 100,000 a year.
Separately, the Home Office has released statistics on the number of British citizenship applications, which show the number of EU nationals applying has increased by almost a third over the past year.
There were 43,545 requests in the 12 months to the end of September - a rise of 32% on last year.