**Solution:**

To solve this problem, we can use Graham's Law of Effusion, which states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. Since the pressure and temperature remain constant, we can assume that the gas behaves ideally and that the molar mass is directly proportional to the mass of the gas.

Let's break down the problem into steps:

**Step 1: Calculate the initial molar mass of helium**
The molar mass of helium (He) is approximately 4 g/mol. Therefore, the initial number of moles of helium in the balloon can be calculated as follows:

Number of moles of helium = Mass of helium / Molar mass of helium
Number of moles of helium = 1 g / 4 g/mol
Number of moles of helium = 0.25 mol

**Step 2: Calculate the final number of moles of helium**
Since 0.27 g of helium effused out of the balloon, the mass of helium remaining in the balloon is:

Remaining mass of helium = Initial mass of helium - Mass of helium effused
Remaining mass of helium = 1 g - 0.27 g
Remaining mass of helium = 0.73 g

Using the molar mass of helium, we can calculate the final number of moles of helium:

Number of moles of helium = Remaining mass of helium / Molar mass of helium
Number of moles of helium = 0.73 g / 4 g/mol
Number of moles of helium = 0.1825 mol

**Step 3: Calculate the final radius of the balloon**
We know that the volume of a balloon is directly proportional to the number of moles of gas it contains. Therefore, we can use the ideal gas law to relate the initial and final volumes of the balloon:

V1 / V2 = n1 / n2

Where:
V1 = Initial volume of the balloon
V2 = Final volume of the balloon
n1 = Initial number of moles of helium
n2 = Final number of moles of helium

Since the radius of the balloon is directly proportional to the cube root of its volume, we can rewrite the equation as:

(r1)^3 / (r2)^3 = n1 / n2

Where:
r1 = Initial radius of the balloon
r2 = Final radius of the balloon

Plugging in the values, we get:

(10 cm)^3 / (r2)^3 = 0.25 mol / 0.1825 mol

Simplifying the equation:

(r2)^3 = (10 cm)^3 * (0.1825 mol / 0.25 mol)

(r2)^3 = (10 cm)^3 * (0.73)

Taking the cube root of both sides:

r2 = (10 cm) * (0.73)^(1/3)

Using a calculator, we find:

r2 ≈ 9.14 cm

Therefore, the radius of the balloon after 0.27 g of helium effused out is approximately 9.14 cm.