5.8 years
According to this page, distance travelled under constant force acceleration, even up to relativistic speeds, is calculated by:
$s(t) = c(\frac{m_0c}{F})(\sqrt{1 + (\frac{F}{m_0c})^2t^2} - 1)$
...and since...
$\frac{F}{m_0} = a = g \approx 10 m/s^2$
...and since we accelerate half-way and decelerate halfway we calculate the time to travel half the distance and then double that. Half the distance is...
$s=300\cdot10^6m/s\cdot60s/min\cdot60min/h\cdot24h/day\cdot365day/y\cdot2.12y \approx 20\cdot10^{15}m$
...so from this we get...
$20\cdot10^{15} = 3\cdot10^8\cdot\frac{3\cdot10^8}{10}\cdot(\sqrt{1 + \frac{10^2}{3\cdot10^8\cdot3\cdot10^8}t^2} - 1)$$20\cdot10^{15} = 3\cdot10^8\cdot\frac{3\cdot10^8}{10}\cdot(\sqrt{1 + \frac{10\cdot10}{3\cdot10^8\cdot3\cdot10^8}t^2} - 1)$
...which gives us...
$t = 92\cdot10^6s = 2.9y$
Double this and you get a total traveltime of 5.8 years.