The spectral gap $\\gamma$ of a finite, ergodic, and reversible Markov chain\nis an important parameter measuring the asymptotic rate of convergence. In\napplications, the transition matrix $P$ may be unknown, yet one sample of the\nchain up to a fixed time $n$ may be observed. We consider here the problem of\nestimating $\\gamma$ from this data. Let $\\pi$ be the stationary distribution of\n$P$, and $\\pi_\\star = \\min_x \\pi(x)$. We show that if $n =\n\\tilde{O}\\bigl(\\frac{1}{\\gamma \\pi_\\star}\\bigr)$, then $\\gamma$ can be\nestimated to within multiplicative constants with high probability. When $\\pi$\nis uniform on $d$ states, this matches (up to logarithmic correction) a lower\nbound of $\\tilde{\\Omega}\\bigl(\\frac{d}{\\gamma}\\bigr)$ steps required for\nprecise estimation of $\\gamma$. Moreover, we provide the first procedure for\ncomputing a fully data-dependent interval, from a single finite-length\ntrajectory of the chain, that traps the mixing time $t_{\\text{mix}}$ of the\nchain at a prescribed confidence level. The interval does not require the\nknowledge of any parameters of the chain. This stands in contrast to previous\napproaches, which either only provide point estimates, or require a reset\nmechanism, or additional prior knowledge. The interval is constructed around\nthe relaxation time $t_{\\text{relax}} = 1/\\gamma$, which is strongly related to\nthe mixing time, and the width of the interval converges to zero roughly at a\n$1/\\sqrt{n}$ rate, where $n$ is the length of the sample path.\n