Simulating first passage times

Question

I have a Brownian motion $X_t = \mu t+\sigma W_t$, where $W_t$ standard Brownian motion. I know that the first passage time $\tau = \min\{t|X_t\leq\alpha\}$, is Inverse Gaussian distributed i.e., $\tau\sim IG(\alpha/\mu,\alpha^2/\sigma^2)$ which I can generate easily using software such as R. I want to generate first passage times in the interval $[0,T]$. The parameters for $IG(\alpha/\mu,\alpha^2/\sigma^2)$ are constants with no dependency on T. My problem is that the values I am getting a too big for an interval such as $[0,1]$.