The bill entered into force immediately after it was signed, Papuashvili announced at a briefing, Echo of the Caucasus reported.
Georgia's parliament had earlier overrode the veto of Georgian President Salome Zurabishvili to officially approve the law.
84 deputies voted for the bill, 4 against.
Authorities must adopt by-laws within two months. Georgia's Justice Ministry and the National Agency of the Public Registry must carry out "preliminary logistical and other measures for the implementation of this law."
The Transparency of Foreign Influence Law, known as the "Foreign Agents" law, or "Russian law", declares non-governmental institutions and media outlets with more than 20% of their income from abroad as organizations representing the interests of a foreign state.
These organizations now must apply to the National Public Registry Agency to register as "foreign agents". They must also fill out a financial declaration every year. Otherwise, they will be fined $8,500 for the first violation.