Biomarker data is often subject to limits of quantification or limits of detection. Statistically, this corresponds to left- or interval-censoring. In applications, e.g. when a biomarker is a covariate in a regression model, such data is often imputed in some way, e.g. by considering the limit of detection an actual measurement. In order to be able to correctly account for the nature of the data, we have implemented maximum likelihood estimation in Weibull regression for a right-censored endpoint, one interval-censored, and an arbitrary number of non-censored covariates. We discuss the assumptions made in the model and how to set up the likelihood function and maximize it. Inference for estimated parameters can be received using standard maximum likelihood theory. We quantify the bias and mean-squared error for parameter estimates compared to commonly used imputation methods. We illustrate the methodology by applying it to assess Prentice´s criteria for surrogacy in data simulated from a randomized clinical registration trial. The software is available on CRAN, as the package SurvRegCensCov.