Show that stochastic integral is Gaussian by independency of increments

Question

Let $h \in L^2[(0,1)]$ and consider the process $(X_t)_{t \in [0,1]} = \big( \int_0^t h(s) \text{d}B_s \big)_{t \in [0,1]}$, where $B_s$ is Brownian motion.

By construction of the integral I know that $(X_t)$ has independent increments. How can I deduce from this fact that $(X_t)$ is Gaussian?

Thanks in advance!