Motivation: Amplicon sequencing is widely applied to explore heterogeneity and rare variants in genetic populations. Resolving true biological variants and quantifying their abundance is crucial for downstream analyses, but measured abundances are distorted by stochasticity and bias in amplification, plus errors during Polymerase Chain Reaction (PCR) and sequencing. One solution attaches Unique Molecular Identifiers (UMIs) to sample sequences before amplification eliminating amplification bias by clustering reads on UMI and counting clusters to quantify abundance. While modern methods improve over naive clustering by UMI identity, most do not account for UMI reuse, or collision, and they do not adequately model PCR and sequencing errors in the UMIs and sample sequences. Results: We introduce Deduplication and accurate Abundance estimation with UMIs (DAUMI), a probabilistic framework to detect true biological sequences and accurately estimate their deduplicated abundance from amplicon sequence data. DAUMI recognizes UMI collision, even on highly similar sequences, and detects and corrects most PCR and sequencing errors in the UMI and sampled sequences. We demonstrate DAUMI performs better on simulated and real data compared to other UMI-aware clustering methods. Availability: Source code is available at https://github.com/xiyupeng/AmpliCI-UMI.