The development, which will be in stages as finance permits, will include the stadium's capacity being increase from the current figure of 10,744 to 20,000.
The Aviva Premiership side's chairman and chief executive Tony Rowe hailed the news from Wednesday's annual general meeting as an "important milestone" in the club's history.
He said: "As a board we are delighted that the membership of Exeter Rugby Club have supported our plans for the future.
"Over the last few years we have made great strides in developing the club, but we are keen to push forward even further and make ourselves a force within not only the English game, but also the European game.
"This decision by the members is yet another important milestone in the history of our great club."
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
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The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.