Does H2CO3 exist in solution?
Aqueous carbonate solutions contain four different solute species:


$\ce{CO2(aq)}$
$\ce{H2CO3}$
$\ce{HCO3-}$
$\ce{CO3^2-}$


The equilibrium for the hydration reaction

$$\ce{H2O + CO2(aq) &lt;=&gt; H2CO3}\tag1$$

lies to the left. The corresponding equilibrium constant $K$ at $25\ \mathrm{^\circ C}$ is about

$$K=\frac{\left[\ce{CO2(aq)}\right]}{\left[\ce{H2CO3}\right]}\approx650$$

(literature values of $K$ range from $350$ to $990$.)

The commonly used first acid dissociation constant of carbonic acid $\mathrm pK_{\mathrm a1}=6.35$ (at $25\ \mathrm{^\circ C}$) actually is a composite constant that includes both the hydration reaction $\text{(1)}$ and the protolysis of true $\ce{H2CO3}$ $\text{(2)}$.

$$\ce{H2CO3 &lt;=&gt; H+ + HCO3-}\tag2$$

The constants $K$, $K_{\mathrm a1}$, and the actual acid dissociation constant $K_{\ce{H2CO3}}$ of true $\ce{H2CO3}$ are interrelated according to

$$K_{\mathrm a1}=\frac{K_{\ce{H2CO3}}}{1+K}\approx\frac{K_{\ce{H2CO3}}}{K}$$

Calculated values for $\mathrm pK_{\ce{H2CO3}}$ range from $3.8$ to $3.4$. Thus, true $\ce{H2CO3}$ is stronger than acetic acid and formic acid.

The second acid dissociation constant of carbonic acid is $\mathrm pK_{\mathrm a2}=10.33$ (at $25\ \mathrm{^\circ C}$).

Therefore, the predominant species in aqueous carbonate solutions (at $25\ \mathrm{^\circ C}$) are


$\ce{CO3^2-}$ at $10.33\lt\mathrm{pH}$
$\ce{HCO3-}$ at $6.35\lt\mathrm{pH}\lt10.33$
$\ce{CO2(aq)}$ and $\ce{H2CO3}$ at $\mathrm{pH}\lt6.35$


Thus, true $\ce{H2CO3}$ primarily exists in acidic solutions. However, because of the hydration equilibrium $\text{(1)}$ and the corresponding equilibrium constant $K\approx650$, most of the true $\ce{H2CO3}$ is converted to $\ce{CO2(aq)}$.

(All values taken from: Stumm, W.; Morgan, J. J. Aquatic Chemistry, Third Edition; John Wiley &amp; Sons: New York, NY, 1996; pp 150–152.)