By John Parry

NEW YORK (Reuters) - The U.S. service sector was still shrinking last month but at a slowing pace, with activity at the highest since September 2008, when Lehman Brothers' collapse exacerbated the global financial crisis, a report showed on Monday.

At the same time, a separate measure of job growth fell slightly in June from May's level.

The Institute for Supply Management said its measure of the service sector rose to 47.0 last month from 44.0 in May. The reading was above economists' median forecast for a rise to 46.0, but it was still not indicative of a definite turnaround.

"It's a good number, not quite showing expansion yet, but rising closer to that 50-level that divides contraction from expansion," said Gary Thayer, senior economist with Wells Fargo Advisors in St. Louis.

The services sector represents about 80 percent of U.S. economic activity, including businesses such as banks, airlines, hotels and restaurants.

Both manufacturing and service sector reports show signs that the 18-month-old U.S. recession, the most protracted in decades, may soon end.

The ISM service sector report's business activity index jumped within a hair of expansion territory, to 49.8, from 42.4 in May.

The new orders index rose to 48.6 in June from 44.4 in May. Prices paid rose to 53.7 in June from 46.9, driven partly by a rise in oil prices, said Anthony Nieves, chair of the ISM non-manufacturing business survey committee.