British wages have picked up in recent months, and figures due on Wednesday are expected to show earnings excluding bonuses to record their biggest rise since early 2009.
However, downward pressures on inflation are likely to persist for some months yet. Brent crude oil prices fell by around $2 a barrel in June, and have dropped further at the start of July.
The ONS said the downward effect from clothing on inflation was particularly marked because June 2014 saw unseasonable rises in prices, whereas this year saw the start of the usual seasonal sales.
Falls in food prices were led by goods such as pizzas, which are commonly used to lead supermarket discounts, capping a year of declining prices -- the longest unbroken run since 2000.
Air fares rose less than last year, particularly for European destinations.
Many economists say the recent rise of sterling against a basket of currencies may limit the pace of future price rises.
Data also released by the ONS on Tuesday showed that factory gate prices fell 1.5 percent in June, the smallest drop so far this year, but as expected by economists.
House prices across Britain rose by 5.7 percent in the 12 months to May, up from 5.5 percent in April, the ONS also said on Tuesday.