Scaling an audio signal to a given rms-in-dB value

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I need to scale a given audio signal to a given acoustic rms-amplification (60dB) and a given base acoustic pressure (p0=20*10^(-6)). How to do this?
What I've tried so far:
RMS_signal_dB = 20*log10(rms(signal)/p0)
% Amplification difference to 60dB
Diff_signal_dB = 60 - RMS_signal_dB;
% Compute factor
Diff_signal = p0 * 10^(Diff_signal_dB/20);
% Scale
new_signal = Diff_signal*signal;
% New value on 60 dB?
new_RMS_signal_dB = 20*log10(rms(new_signal)/p0);
The new rms amplification value differs from 60 dB... Could be a mathematical fault, maybe someone can help. Thanks!

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