The announcement came as the Glasgow-based group also reported of a further sharp fall in profits.
The slump in the oil and gas sector led to a 25% drop in pre-tax earnings for January to June, to £82m. Revenue was down 12% to £866m.
Mr Cochrane said there were improving signs from sales and servicing of equipment for the mining sector.
Weir was also expected to gain from the go-ahead for the Hinkley Point nuclear power station, as one of the suppliers of equipment.
The company's share price fell by 4% following the results announcement.
The board announced its finance director, Jon Stanton, will be next chief executive, taking over on 1 October.
The long-established Glasgow firm has cut 490 jobs in the first half of the year, 90 more than it had announced in November. It has shed half of its American workforce.
The chief executive said the results were ahead of expectations.
He said that while the oil and gas sector was difficult Weir grew its market share, "aggressively" cutting costs.
Mr Cochrane said it could benefit, by the end of this year, from the conversion of foreign earnings into weakened sterling.
He said: "While there have been early signs of stability in oil and gas markets, activity remains at very low levels with only a modest improvement expected in late 2016.
"Conversely, there are signs of mining markets normalising, supported by commodity prices and an improving pipeline of brownfield opportunities".
Engineering giant Weir Group has announced its chief executive, Keith Cochrane, is to step down.