The 27-year-old New Zealand-born back, now qualified to represent Ireland through residency, previously played for New Zealand at Schools, Under-19 and Under-21 levels.
Connacht Chief Executive Tom Sears said: "Miah is a key member of the Connacht squad and we are very pleased he will be staying with us.
"Every squad needs two quality players at out-half and Miah has proven time and time again that when he gets the opportunity he is more than capable of stepping up. He showed that admirably last week in the win over the Cardiff Blues.
"He has an outstanding attitude and work ethic and sets a great example to other players.
"With both Miah and Dan Parks in our squad we have two excellent goal-kicking 10s who push each other constantly and provide crucial depth in a key position."