No, we attempt to prove it and sometimes fail to do so, at which point we are justified in rejecting the null.

Perhaps prove/disprove is a misleading choice of words, but with those terms, the null hypothesis is never proved.

When the null hypothesis is rejected, it is because our test results are below some very small threshold of probability given the assumption of the null (this is then taken as "proving" the alternative hypothesis for the purposes of this test). When the null is not rejected, our results might be just above that threshold, which by our method doesn't reject the null (and neither proves nor disproves the alternative) but certainly doesn't prove the null either. We never prove the null hypothesis: if that's what we were trying to do it wouldn't be the null.