AC-coupled amplifiers are widely used in analog signal processing, particularly in audio and biopotential amplifiers. Applications that require very low cutoff frequencies 𝒇𝑪 lead to large time constants 𝝉 = 𝑹𝑪, thus resulting in large 𝑹 values because the capacitance of high-quality non-polarized capacitors is limited to a few 𝝁𝑭. Moreover, in a classic approach using a passive 𝑹𝑪 network before the amplifier, the noise of this high value resistor forces to set 𝒇𝑪 well below the bandwidth of interest to reduce its noise contribution, thus leading to very long transient time responses. The presented analysis shows that the use of DC servo loop circuits (DCSL) allows becoming negligible the noise contribution of the resistor, that works as a “cold resistor”: it behaves as a resistor of value 𝑹 to set 𝒇𝑪 but with a noise significantly lower than its theoretical thermal noise. The inverting and non-inverting DCSL topologies were analyzed showing that the cost to pay for this noise reduction is a limitation on the DC voltages the DCSLs admit at their inputs. The non-inverting scheme allows for achieving the lowest noise but the lowest DC input range. The circuits' analytic noise transfer functions and their experimental validation are provided. As an example, a DCSL optimized to achieve a very-low voltage noise, a gain of 40 dB and 𝒇𝑪 = 𝟏. 𝟔 𝐇𝐳 was designed, built, and tested showing that its input referred noise 𝒆𝒊 is exclusively due to the input operational amplifier. Using an OPA221 of Texas InstrumentsTM the DCSL featured 𝒆𝒊=𝟏. 𝟐 𝐧𝐕⁄√𝐇𝐳 @𝟏 𝐤𝐇𝐳.