We prove that every language in $\mathrm{P}^{\# \mathrm{P}}$ can be decided by a bounded-error randomized algorithm that uses only $O(\log n)$ bits of work space. The algorithm is guaranteed to halt for every input and every setting of the random tape. However, there is a catch: the algorithm uses a *multi-access* random tape, i.e., the algorithm scans both forward and backward across a read-only tape filled with an unlimited number of random bits. Prior work on log-space algorithms with multi-access randomness either focuses on polynomial-time algorithms, or else permits algorithms that sometimes run forever. Our algorithm always halts, but it uses exponential time. Thus, our work identifies a natural model of computation in which randomness is intrinsically useful, assuming $\mathrm{L} \neq \mathrm{P}^{\# \mathrm{P}}$.