- 2015 Society for Industrial and Applied Mathematics. A classical inverse problem is "can you hear the density of a string clamped at both ends"? The answer as narrowly stated is negative; one has to give an additional data sequence such as the energy in each mode (norming constants) or information on the string slope at the endpoints or, alternatively, change the boundary conditions and obtain the new frequencies. What if neither of these options is possible? One solution recently proposed was to add a known mass-density and remeasure the Dirichlet spectrum [W. Rundell and P. E. Sacks, SIAM J. Appl. Math., 73 (2013), pp. 1020-1037]. In this work we take a related approach: we are able to add any number of known mass-densities but we can only measure the fundamental frequency, that is, the lowest eigenvalue, in each case. We will show that our problem can be closely approximated by a related problem for which there is uniqueness and we discuss a suite of reconstruction methods.