Numerical Integration of a high dimensional function

Question

My problem is the following, I have a function, with an input vector of $\mathbb{R^{3072}}$, which outputs {$\mathbb{ x \in R : 0 \leq x \leq 1}$}. I want to find the integral of the function over the box $\mathbb{[0, 255]^{3072}}$. I know about the function, that in most case it outputs $\mathbb{0}$ or a value close to $\mathbb{0}$ and I know some vectors for which it outputs a value close to $\mathbb{1}$. I so far tried to use crude Monte Carlo Integration in order to approximate the integral, however the results are not very good, due to the very low probability of getting a random point that results in an output close to $\mathbb{1}$. I therefore wanted to ask if someone has an idea how to tackle my problem in order to find a better approximation of the integral of my function, given the available information about the function.